Logarithmic Functions - AP Precalculus

1. Since $10^2 = 100$, the answer is 2.

1. Convert to exponential: $3^4 = 81$.

1. Since $7^2 = 49$, the answer is 2.

Undefined. Logarithm of zero does not exist in real numbers.

1. Logarithm and exponential with same base cancel out.

x. Logarithm and exponential with same base cancel out.

1. Natural log and $e$ with same exponent cancel out.

1. Logarithm and exponential with same base cancel out.

$5^3 = 125$. Direct conversion from logarithmic to exponential form.

1. Any base raised to power 1 equals itself.

$n \times \text{log}_b(a)$. Power rule moves exponent $n$ as coefficient.

1. Since $b^0 = 1$ and $\text{log}_b(1) = 0$.

-1. Quotient simplifies to $b^{-1}$, so $\text{log}_b(b^{-1}) = -1$.

1. Natural log of its own base equals 1.

1. Since $7^2 = 49$, the answer is 2.

$-\text{log}_b(x)$. Reciprocal creates negative exponent, so negative log.

1. Natural log of 1 equals 0 since $e^0 = 1$.

$\text{log}_b(x) - \text{log}_b(y)$. Quotient rule expands division into subtraction of logs.

x. Exponential and natural log are inverse functions.

-1. Since $10^{-1} = 0.1$, the answer is -1.

1. Logarithm and exponential with same base cancel out.

All real numbers. Logarithms can output any real value as input approaches 0 or infinity.

$3 \times \text{log}_b(x)$. Power rule moves the exponent as a coefficient.

1. Any base raised to the power 0 equals 1.

1. Since $4^3 = 64$, the answer is 3.

1. Natural log and $e$ with same exponent cancel out.

$\text{log}_{10}(1000) = x$. Direct conversion from exponential to logarithmic form.

x. Logarithm and exponential with same base cancel out.

1. Any base raised to the power 0 equals 1.