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log_(7)48

$6$ . $\log _{7} 48$

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Evaluate logarithms

Full solution

Q.

6

.

\log _{7} 48

Understand the problem: Understand the problem. $\newline$ We need to find the value of the logarithm of $48$ with base $7$ , which is written as $\log_7(48)$ .
Express as power of $7$ : Express $48$ as a power of $7$ , if possible. $\newline$ $48$ is not an integer power of $7$ , so we cannot directly find an integer answer like in the previous example. We will need to use a calculator or logarithm properties to approximate the value or express it in terms of logarithms we know.
Use logarithm properties: Use logarithm properties to simplify the expression, if possible. $\newline$ Since $48$ is not a power of $7$ , we cannot simplify $\log_7(48)$ to an integer. However, we can use the change of base formula to express this logarithm in terms of natural logarithms ( $\ln$ ) or common logarithms ( $\log$ ), which are more easily calculated with a calculator. $\newline$ The change of base formula is: $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$ , where $c$ is a new base we choose.
Apply change of base: Apply the change of base formula. $\newline$ Let's use the natural logarithm (base $e$ ). We can rewrite $\log_7(48)$ as $\ln(48) / \ln(7)$ .
Calculate with calculator: Calculate the value using a calculator. $\newline$ Using a calculator, we find: $\newline$ $\ln(48) \approx 3.8712$ $\newline$ $\ln(7) \approx 1.9459$ $\newline$ Now, divide $\ln(48)$ by $\ln(7)$ to get the value of $\log_7(48)$ . $\newline$ $\log_7(48) \approx \frac{3.8712}{1.9459} \approx 1.9890$

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