Logarithm

Property	Equation
Product Rule	logₐ(x * y) = logₐ(x) + logₐ(y)
Quotient Rule	logₐ(x / y) = logₐ(x) – logₐ(y)
Power Rule	logₐ(x^k) = k * logₐ(x)
Change of Base Rule	logₐ(x) = logᵦ(x) / logᵦ(a)
Inverse of Exponential	logₐ(a^x) = x
Logarithm of 1	logₐ(1) = 0
Logarithm of Base	logₐ(a) = 1
Logarithm of Zero	logₐ(0) is undefined (not defined)
Logarithm of Negative	logₐ(x) is undefined for x ≤ 0
Logarithm of Negative Base	log₋ₐ(x) is undefined for x > 0 and a < 1

Solve for x: log2(x) = 4.
Solution: Since the base is 2, rewrite the equation as x = 2^4 = 16.
Requested by: Ali

Solve for x: log5(x) = 2.
Solution: Rewrite the equation as x = 5^2 = 25.
Requested by: Ali

Simplify: log3(9) + log3(27).
**Solution: Use the logarithmic property loga(b) + loga(c) = loga(b * c).
log3(9) + log3(27) = log3(9 * 27) = log3(243).
Requested by: Mazhar**

Simplify: log6(36) – log6(6).
Solution: Use the logarithmic property loga(b) – loga(c) = loga(b / c).
log6(36) – log6(6) = log6(36 / 6) = log6(6).
Requested by: Mazhar

Solve for x: 2^(3x – 1) = 8.
Solution: Rewrite 8 as a power of 2: 8 = 2^3.
2^(3x – 1) = 2^3.
Equate the exponents: 3x – 1 = 3.
Solve for x: 3x = 4, x = 4/3.
Requested by: Zain

Simplify: log4(64) / log4(16).
Solution: Use the change of base formula: loga(b) / loga(c) = logc(b).
log4(64) / log4(16) = log16(64).
Requested by: Zain

Solve for x: log3(x + 2) + log3(x – 1) = 2.
**Solution: Combine the logarithms using the logarithmic property loga(b) + loga(c) = loga(b * c).
log3((x + 2)(x – 1)) = 2.
Rewrite the equation in exponential form: 3^2 = (x + 2)(x – 1).
Solve for x: 9 = x^2 + x – 2, x^2 + x – 11 = 0.
Factor or use the quadratic formula to find x.
Requested by: Ali**

Simplify: log2(8) + log3(9) – log4(16).
Solution: Use the logarithmic properties and simplify each term individually.
log2(8) + log3(9) – log4(16) = log2(2^3) + log3(3^2) – log4(4^2).
= 3log2(2) + 2log3(3) – 2log4(4).
= 3(1) + 2(1) – 2(2).
= 3 + 2 – 4.
= 1.
Requested by: Zain

Solve for x: logx(1/100) = -2.
Solution: Rewrite the equation as x^(-2) = 1/100.
Take the reciprocal of both sides: x^2 = 100.
Solve for x: x = ±10.
Requested by: Daniyal

Simplify: log5(25^x).
**Solution: Use the logarithmic property loga(b^c) = c * loga(b).
log5(25^x) = x * log5(25).
Since 25 = 5^2, log5(25) = 2.
Simplify: x * 2 = 2x.
Requested by: Ali**

Solve for x: 3^(2x + 1) = 27.
Solution: Rewrite 27 as a power of 3: 27 = 3^3.
3^(2x + 1) = 3^3.
Equate the exponents: 2x + 1 = 3.
Solve for x: 2x = 2, x = 1.
Requested by: Ali

Simplify: log7(49) – log7(7^2).
Solution: Use the logarithmic property loga(b) – loga(c) = loga(b / c).
log7(49) – log7(7^2) = log7(49 / 7^2) = log7(1) = 0.
Requested by: Zain

Solve for x: log(x – 2) = log(x + 3) – 2.
Solution: Combine the logarithms using the logarithmic property loga(b) – loga(c) = loga(b / c).
log(x – 2) = log(x + 3) – 2.
Rewrite the equation in exponential form: x – 2 = (x + 3) / 100.
Solve for x: x = 102/99.
Requested by: Aakash

Simplify: log2(16) – log2(2).
Solution: Use the logarithmic property loga(b) – loga(c) = loga(b / c).
log2(16) – log2(2) = log2(16 / 2) = log2(8).
Requested by: Arslan

Solve for x: 10^(2x – 3) = 100.
Solution: Rewrite 100 as a power of 10: 100 = 10^2.
10^(2x – 3) = 10^2.
Equate the exponents: 2x – 3 = 2.
Solve for x: 2x = 5, x = 5/2.
Requested by: Arslan

Simplify: log4(8) – log4(2).
Solution: Use the logarithmic property loga(b) – loga(c) = loga(b / c).
log4(8) – log4(2) = log4(8 / 2) = log4(4).
Requested by: Ali

Solve for x: log5(x^2) = 3.
Solution: Rewrite the equation as x^2 = 5^3.
Solve for x: x = ±√(5^3).
Requested by: Ali

Simplify: log3(81) – log3(9) + log3(27).
Solution: Use the logarithmic properties and simplify each term individually.
log3(81) – log3(9) + log3(27) = log3(3^4) – log3(3^2) + log3(3^3).
= 4log3(3) – 2log3(3) + 3log3(3).
= 4(1) – 2(1) + 3(1).
= 4 – 2 + 3.
= 5.
Requested by: Zain

Solve for x: 2^(x + 1) = 8.
Solution: Rewrite 8 as a power of 2: 8 = 2^3.
2^(x + 1) = 2^3.
Equate the exponents: x + 1 = 3.
Solve for x: x = 2.
Requested by: Ali

Simplify: log6(36) + log6(6) – log6(6^2).
Solution: Use the logarithmic properties and simplify each term individually.
log6(36) + log6(6) – log6(6^2) = log6(6^2) + log6(6) – log6(6^2).
= 2log6(6) + log6(6) – 2log6(6).
= 2(1) + 1 – 2(1).
= 2 + 1 – 2.
= 1
Requested by Zain

Solve for x: log2(x) = 4. Solution: Since the base is 2, rewrite the equation as x = 2^4 = 16. Requested by: Ali

Solve for x: log5(x) = 2. Solution: Rewrite the equation as x = 5^2 = 25. Requested by: Ali

Simplify: log3(9) + log3(27). Solution: Use the logarithmic property loga(b) + loga(c) = loga(b * c). log3(9) + log3(27) = log3(9 * 27) = log3(243). Requested by: Mazhar

Simplify: log6(36) – log6(6). Solution: Use the logarithmic property loga(b) – loga(c) = loga(b / c). log6(36) – log6(6) = log6(36 / 6) = log6(6). Requested by: Mazhar

Solve for x: 2^(3x – 1) = 8. Solution: Rewrite 8 as a power of 2: 8 = 2^3. 2^(3x – 1) = 2^3. Equate the exponents: 3x – 1 = 3. Solve for x: 3x = 4, x = 4/3. Requested by: Zain

Simplify: log4(64) / log4(16). Solution: Use the change of base formula: loga(b) / loga(c) = logc(b). log4(64) / log4(16) = log16(64). Requested by: Zain

Simplify: log2(8) + log3(9) – log4(16). Solution: Use the logarithmic properties and simplify each term individually. log2(8) + log3(9) – log4(16) = log2(2^3) + log3(3^2) – log4(4^2). = 3log2(2) + 2log3(3) – 2log4(4). = 3(1) + 2(1) – 2(2). = 3 + 2 – 4. = 1. Requested by: Zain

Solve for x: logx(1/100) = -2. Solution: Rewrite the equation as x^(-2) = 1/100. Take the reciprocal of both sides: x^2 = 100. Solve for x: x = ±10. Requested by: Daniyal

Simplify: log5(25^x). Solution: Use the logarithmic property loga(b^c) = c * loga(b). log5(25^x) = x * log5(25). Since 25 = 5^2, log5(25) = 2. Simplify: x * 2 = 2x. Requested by: Ali

Solve for x: 3^(2x + 1) = 27. Solution: Rewrite 27 as a power of 3: 27 = 3^3. 3^(2x + 1) = 3^3. Equate the exponents: 2x + 1 = 3. Solve for x: 2x = 2, x = 1. Requested by: Ali

Simplify: log7(49) – log7(7^2). Solution: Use the logarithmic property loga(b) – loga(c) = loga(b / c). log7(49) – log7(7^2) = log7(49 / 7^2) = log7(1) = 0. Requested by: Zain

Solve for x: log(x – 2) = log(x + 3) – 2. Solution: Combine the logarithms using the logarithmic property loga(b) – loga(c) = loga(b / c). log(x – 2) = log(x + 3) – 2. Rewrite the equation in exponential form: x – 2 = (x + 3) / 100. Solve for x: x = 102/99. Requested by: Aakash

Simplify: log2(16) – log2(2). Solution: Use the logarithmic property loga(b) – loga(c) = loga(b / c). log2(16) – log2(2) = log2(16 / 2) = log2(8). Requested by: Arslan

Solve for x: 10^(2x – 3) = 100. Solution: Rewrite 100 as a power of 10: 100 = 10^2. 10^(2x – 3) = 10^2. Equate the exponents: 2x – 3 = 2. Solve for x: 2x = 5, x = 5/2. Requested by: Arslan

Simplify: log4(8) – log4(2). Solution: Use the logarithmic property loga(b) – loga(c) = loga(b / c). log4(8) – log4(2) = log4(8 / 2) = log4(4). Requested by: Ali

Solve for x: log5(x^2) = 3. Solution: Rewrite the equation as x^2 = 5^3. Solve for x: x = ±√(5^3). Requested by: Ali

Simplify: log3(81) – log3(9) + log3(27). Solution: Use the logarithmic properties and simplify each term individually. log3(81) – log3(9) + log3(27) = log3(3^4) – log3(3^2) + log3(3^3). = 4log3(3) – 2log3(3) + 3log3(3). = 4(1) – 2(1) + 3(1). = 4 – 2 + 3. = 5. Requested by: Zain

Solve for x: 2^(x + 1) = 8. Solution: Rewrite 8 as a power of 2: 8 = 2^3. 2^(x + 1) = 2^3. Equate the exponents: x + 1 = 3. Solve for x: x = 2. Requested by: Ali

Simplify: log6(36) + log6(6) – log6(6^2). Solution: Use the logarithmic properties and simplify each term individually. log6(36) + log6(6) – log6(6^2) = log6(6^2) + log6(6) – log6(6^2). = 2log6(6) + log6(6) – 2log6(6). = 2(1) + 1 – 2(1). = 2 + 1 – 2. = 1 Requested by Zain

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Solve for x: log2(x) = 4.
Solution: Since the base is 2, rewrite the equation as x = 2^4 = 16.
Requested by: Ali

Solve for x: log5(x) = 2.
Solution: Rewrite the equation as x = 5^2 = 25.
Requested by: Ali

Simplify: log3(9) + log3(27).
**Solution: Use the logarithmic property loga(b) + loga(c) = loga(b * c).
log3(9) + log3(27) = log3(9 * 27) = log3(243).
Requested by: Mazhar**

Simplify: log6(36) – log6(6).
Solution: Use the logarithmic property loga(b) – loga(c) = loga(b / c).
log6(36) – log6(6) = log6(36 / 6) = log6(6).
Requested by: Mazhar

Solve for x: 2^(3x – 1) = 8.
Solution: Rewrite 8 as a power of 2: 8 = 2^3.
2^(3x – 1) = 2^3.
Equate the exponents: 3x – 1 = 3.
Solve for x: 3x = 4, x = 4/3.
Requested by: Zain

Simplify: log4(64) / log4(16).
Solution: Use the change of base formula: loga(b) / loga(c) = logc(b).
log4(64) / log4(16) = log16(64).
Requested by: Zain

Simplify: log2(8) + log3(9) – log4(16).
Solution: Use the logarithmic properties and simplify each term individually.
log2(8) + log3(9) – log4(16) = log2(2^3) + log3(3^2) – log4(4^2).
= 3log2(2) + 2log3(3) – 2log4(4).
= 3(1) + 2(1) – 2(2).
= 3 + 2 – 4.
= 1.
Requested by: Zain

Solve for x: logx(1/100) = -2.
Solution: Rewrite the equation as x^(-2) = 1/100.
Take the reciprocal of both sides: x^2 = 100.
Solve for x: x = ±10.
Requested by: Daniyal

Simplify: log5(25^x).
**Solution: Use the logarithmic property loga(b^c) = c * loga(b).
log5(25^x) = x * log5(25).
Since 25 = 5^2, log5(25) = 2.
Simplify: x * 2 = 2x.
Requested by: Ali**

Solve for x: 3^(2x + 1) = 27.
Solution: Rewrite 27 as a power of 3: 27 = 3^3.
3^(2x + 1) = 3^3.
Equate the exponents: 2x + 1 = 3.
Solve for x: 2x = 2, x = 1.
Requested by: Ali

Simplify: log7(49) – log7(7^2).
Solution: Use the logarithmic property loga(b) – loga(c) = loga(b / c).
log7(49) – log7(7^2) = log7(49 / 7^2) = log7(1) = 0.
Requested by: Zain

Solve for x: log(x – 2) = log(x + 3) – 2.
Solution: Combine the logarithms using the logarithmic property loga(b) – loga(c) = loga(b / c).
log(x – 2) = log(x + 3) – 2.
Rewrite the equation in exponential form: x – 2 = (x + 3) / 100.
Solve for x: x = 102/99.
Requested by: Aakash

Simplify: log2(16) – log2(2).
Solution: Use the logarithmic property loga(b) – loga(c) = loga(b / c).
log2(16) – log2(2) = log2(16 / 2) = log2(8).
Requested by: Arslan

Solve for x: 10^(2x – 3) = 100.
Solution: Rewrite 100 as a power of 10: 100 = 10^2.
10^(2x – 3) = 10^2.
Equate the exponents: 2x – 3 = 2.
Solve for x: 2x = 5, x = 5/2.
Requested by: Arslan

Simplify: log4(8) – log4(2).
Solution: Use the logarithmic property loga(b) – loga(c) = loga(b / c).
log4(8) – log4(2) = log4(8 / 2) = log4(4).
Requested by: Ali

Solve for x: log5(x^2) = 3.
Solution: Rewrite the equation as x^2 = 5^3.
Solve for x: x = ±√(5^3).
Requested by: Ali

Simplify: log3(81) – log3(9) + log3(27).
Solution: Use the logarithmic properties and simplify each term individually.
log3(81) – log3(9) + log3(27) = log3(3^4) – log3(3^2) + log3(3^3).
= 4log3(3) – 2log3(3) + 3log3(3).
= 4(1) – 2(1) + 3(1).
= 4 – 2 + 3.
= 5.
Requested by: Zain

Solve for x: 2^(x + 1) = 8.
Solution: Rewrite 8 as a power of 2: 8 = 2^3.
2^(x + 1) = 2^3.
Equate the exponents: x + 1 = 3.
Solve for x: x = 2.
Requested by: Ali

Simplify: log6(36) + log6(6) – log6(6^2).
Solution: Use the logarithmic properties and simplify each term individually.
log6(36) + log6(6) – log6(6^2) = log6(6^2) + log6(6) – log6(6^2).
= 2log6(6) + log6(6) – 2log6(6).
= 2(1) + 1 – 2(1).
= 2 + 1 – 2.
= 1
Requested by Zain