Exponential & Logarithmic Intro

Equivalent logarithmic form of the general exponential equation $b^y = x$ (for $b > 0$, $b \neq 1$, and $x > 0$)

$$\log_{b}(x) = y$$

Equivalent exponential form of the logarithmic equation $\log_{b}(x) = y$

$$b^y = x$$

Equivalent exponential form of the natural logarithmic equation $\ln(x) = y$

$$e^y = x$$

Base of the common logarithm $\log(x)$ when no base is explicitly written

Base $10$ (meaning $\log_{10}(x)$)

One-to-one property of exponential expressions: conclusion when $b^u = b^v$ (with $b > 0$, $b \neq 1$)

$u = v$ If the bases are the same, then the exponents must also be equal.

One-to-one property of logarithmic expressions: conclusion when $\log_{b}(u) = \log_{b}(v)$ (with $u, v > 0$)

$u = v$ when the logarithm has the same base.

Solution for $x$ in the basic exponential equation $2^{x+1} = 16$

$$x = 3$$

Solution for $x$ in the basic logarithmic equation $\log_{3}(2x - 1) = 2$

$$x = 5$$