Guided Example #1 - Logarithmic Equations

The equation

$LaTeX: \log_416=2$ $\log_416=2$

tells us that the exponent is 2, the base is 4 and the value is 16.

$LaTeX: \log_416=2\:\longleftrightarrow\:4^2=16$ $\log_416=2\:\longleftrightarrow\:4^2=16$

In general,

$LaTeX: \log_bc=a\:\longleftrightarrow\:b^a=c\left(b>0,\:b\ne a\right)$ $\log_bc=a\:\longleftrightarrow\:b^a=c\left(b>0,\:b\ne a\right)$

$LaTeX: \log_{base}\left(power\right)=exponent$ $\log_{base}\left(power\right)=exponent$

A power is something of the form r for any real number r, and it is called the “rth power of x.”

Write LaTeX: 3^2=9 $3^2=9$ in logarithmic form.

2 is the exponent.

The base is 3.

And the power is 9.

So we write $\log_{3} 9 = 2$ $\log_39=2$

Write $LaTeX: \log_a\left(\frac{1}{a}\right)$

$\log_a\left(\frac{1}{a}\right)$ in exponential form.

Since

\log_{b} c = a \leftrightarrow c (b > 0 a n d b \neq 1)

$\log_bc=a\:\leftrightarrow\:c\left(b>0\:and\:b\ne1\right)$ , we know that a is the base, -1 is the exponent, and

\frac{1}{a}

$\frac{1}{a}$ is the power, therefore the exponential form is

a^{- 1} = \frac{1}{a}

$a^{-1}=\frac{1}{a}$

Solve for x. $LaTeX: \log_5x=2$

$\log_5x=2$

$\log_{5} x = 2$

Rewrite

\log_{5} x = 2

$\log_5x=2$ in exponential form:

5^{2} = x ⟺ x = 25

$5^2=x\:\Longleftrightarrow\:x=25$

Solve for x. $LaTeX: \log_x27=3$

$\log_x27=3$

$\log_{x} 27 = 3$

Rewrite

\log_{x} 27 = 3

$\log_x27=3$ in exponential form:

x^{3} = 27 ⟺ \sqrt[3]{x^{3}} = \sqrt[3]{27} ⟺ x = 3

$x^3=27\:\Longleftrightarrow\:\sqrt[3]{x^3}=\sqrt[3]{27}\:\Longleftrightarrow\:x=3$

\log_{5} x = 2

Rewrite

\log_{2} 16

$\log_216$ in exponential form:

2^{x} = 16 ⟺ x = 4

$2^x=16\:\Longleftrightarrow\:x=4$