QuestionRewrite each expression as an equivalent ratio of logs using the indicated base. $\log _{7}(14)$ to base $e$ . $\square$ Question Help: Written Example Submit Question

Studdy Solution

STEP 1

What is this asking? Rewrite $\log_7(14)$ using base $e$ instead of base 7. Watch out! Remember the change of base formula and don't mix up the different parts of the logarithm!

STEP 2

1. Recall the change of base formula.
2. Apply the change of base formula.

STEP 3

The change of base formula is super handy!
It lets us switch from one base to another.
It says: $\log_a(b) = \frac{\log_c(b)}{\log_c(a)}$ Where $a$ is the original base, $b$ is the argument, and $c$ is the new base we want.

STEP 4

In our problem, we have $\log_7(14)$ , so our original base $a$ is 7, and our argument $b$ is 14.
We want to change to base $e$ , so our new base $c$ is e.
Let's plug these values into our formula: $\log_7(14) = \frac{\log_e(14)}{\log_e(14)}$ Remember, $\log_e(x)$ is the same as $\ln(x)$ , the natural logarithm.
So we can rewrite our answer as: $\log_7(14) = \frac{\ln(14)}{\ln(7)}$ Boom! We've successfully changed the base!

STEP 5

$\frac{\ln(14)}{\ln(7)}$

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