Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Jan 21, 2024

Find the equivalent logarithmic equation for the exponential equation $12=5^{x}$ .

STEP 1

Assumptions
1. We are given an exponential equation: $12 = 5^x$ .
2. We need to find the equivalent logarithmic form of the given exponential equation.
3. We recall the definition of a logarithm: if $a^b = c$ , then $\log_a c = b$ .

STEP 2

Use the definition of a logarithm to convert the exponential equation into its logarithmic form.
According to the definition, if $a^b = c$ , then $\log_a c = b$ . Here, $a$ is the base of the logarithm, $b$ is the logarithm of the number $c$ with respect to base $a$ .

STEP 3

Identify the base ( $a$ ), the exponent ( $b$ ), and the result ( $c$ ) in the given exponential equation.
In the equation $12 = 5^x$ : - The base ( $a$ ) is 5. - The exponent ( $b$ ) is $x$ . - The result ( $c$ ) is 12.

STEP 4

Apply the definition of the logarithm to write the equivalent logarithmic equation.
Using the definition from STEP_2, we have: $\log_a c = b$ Substituting $a = 5$ , $b = x$ , and $c = 12$ , we get: $\log_5 12 = x$

STEP 5

Match the resulting logarithmic equation with the correct answer.
The correct logarithmic equation is: $x = \log_5 12$ This matches with option C.
Therefore, the correct answer is C. $x = \log_5 12$ .

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