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Logarithmic Function Solver

Solve log_a(x)=b and evaluate logarithmic functions

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Solve log_a(x) = b for x
log_(x) =

Logarithmic Function Properties

log_a(x) = b means x = a^b
Domain: x>0, base a>0, a!=1
log(ab)=log(a)+log(b), log(a^n)=n*log(a)
Change of base: log_a(b) = ln(b)/ln(a)

Logarithms are the inverse of exponentials. To solve log_a(x)=b, convert to exponential form: x = a^b. The argument of a log must always be positive. Use log properties to combine or expand log expressions before solving.

⚠ The argument of a logarithm must be positive. Always check your solution by plugging back into the original equation.

What Are Logarithmic Functions?

The logarithm log_a(x) answers: what exponent makes a equal x? f(x)=log_a(x) is the inverse of f(x)=a^x. Domain (0,inf), range (-inf,inf). Natural log ln(x) uses base e. Common log log(x) uses base 10.

Definition

log_a(x) = y means a^y = x. Inverse of exponential. Domain x>0. Range all reals. Asymptote at x=0.

Properties

log(ab)=log(a)+log(b), log(a/b)=log(a)-log(b), log(a^n)=n*log(a). log_a(a)=1, log_a(1)=0.

Solving Method

Convert to exponential: log_a(x)=b -> x=a^b. For expression logs, isolate the log first, then convert.

Natural Log

ln(x) = log_e(x). e=2.71828. The natural log is the inverse of e^x. Used in calculus and science.

Teaching Example: Solve log_2(x)=3. Convert to exponential: x = 2^3 = 8. Check: log_2(8)=3. Domain: 8>0. Verified.

Applications

pH Scale Richter Scale Sound Decibels Half-Life Information Theory

Frequently Asked Questions

Solve log_a(x)=b?▼
Convert to exponential: x = a^b. That is the direct solution. Always check x>0 for domain.
Log properties?▼
log(ab)=log(a)+log(b), log(a/b)=log(a)-log(b), log(a^n)=n*log(a). log_a(a)=1, log_a(1)=0.
Log domain?▼
x>0 for real logs. Base must be >0 and !=1. Asymptote at x=0. Range: all real numbers.
Log of negative?▼
log of a negative number is undefined in the real number system. It exists in complex numbers.

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