Logarithmic Function Manipulation - AP Precalculus

$x = \text{log}_{10}(100)$. Ask: what power of 10 gives 100?

1. Any base raised to power 0 equals 1.

1. Any base to the first power equals itself.

$x = 81$. Convert to exponential form: $3^4 = 81$.

1. Base 10 logarithms are called common logarithms.

$x = 125$. Convert to exponential: $5^3 = 125$.

1. By definition, $\ln(e) = 1$.

1. Since $4^2 = 16$.

$e$. Natural logarithms use Euler's number as base.

1. Since $3^3 = 27$.

1. The inverse property: $\text{log}_b(b^x) = x$.

1. Since $3^3 = 27$.

$\text{log}_b(x^n) = n \times \text{log}_b(x)$. The exponent comes down as a coefficient.

Undefined. Cannot take logarithm of zero or negative numbers.

1. Since $2^5 = 32$.

$\text{log}_b\bigg(\frac{x}{y}\bigg)$. Use the quotient rule in reverse.

1. Since $3^3 = 27$.

If $b^y = x$, then $\text{log}_b(x) = y$. A logarithm asks: what power of $b$ gives $x$?

$\text{log}_{10}(1000) = 3$. Convert by asking: what power of 10 gives 1000?

$\text{log}_b(x^n) = n \times \text{log}_b(x)$. The exponent comes down as a coefficient.

$2^5 = 32$. Convert by writing as $b^y = x$ form.

$\text{log}_b(a) = \frac{\text{log}_c(a)}{\text{log}_c(b)}$. Allows conversion between different logarithm bases.

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

$y = b^x$. Logarithm and exponential functions are inverses.

$\text{log}_b(xy) = \text{log}_b(x) + \text{log}_b(y)$. Addition corresponds to multiplication inside logs.

$\text{log}_b\bigg(\frac{x}{y}\bigg) = \text{log}_b(x) - \text{log}_b(y)$. Subtraction corresponds to division inside logs.

1. The inverse property: $\text{log}_b(b^x) = x$.

$x = 81$. Convert to exponential form: $3^4 = 81$.

1. Any base raised to power 0 equals 1.