Evaluate log base 7 of 25* log base 0.2 of 49

$\log_{7} (25) \cdot \log_{0.2} (49)$

Step 1

Rewrite $\log_{7} (25)$ using the change of base formula.

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{0.2} (49)$

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (25)}{\log (7)} \log_{0.2} (49)$

Step 2

Rewrite $\log_{0.2} (49)$ using the change of base formula.

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\frac{\log (25)}{\log (7)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (25)}{\log (7)} \cdot \frac{\log (49)}{\log (0.2)}$

Step 3

Log base $10$ of $0.2$ is approximately $- 0.69897$ .

$\frac{\log (25)}{\log (7)} \cdot \frac{\log (49)}{- 0.69897}$

Step 4

Move the negative in front of the fraction.

$\frac{\log (25)}{\log (7)} (- \frac{\log (49)}{0.69897})$

Step 5

Log base $10$ of $49$ is approximately $1.69019608$ .

$\frac{\log (25)}{\log (7)} (- \frac{1.69019608}{0.69897})$

Step 6

Divide $1.69019608$ by $0.69897$ .

$\frac{\log (25)}{\log (7)} (- 1 \cdot 2.41812391)$

Step 7

Multiply $- 1$ by $2.41812391$ .

$\frac{\log (25)}{\log (7)} \cdot - 2.41812391$

Step 8

Multiply $\frac{\log (25)}{\log (7)} \cdot - 2.41812391$ .

Combine $\frac{\log (25)}{\log (7)}$ and $- 2.41812391$ .

$\frac{\log (25) \cdot - 2.41812391}{\log (7)}$

Reorder $\log (25)$ and $- 2.41812391$ .

$\frac{- 2.41812391 \cdot \log (25)}{\log (7)}$

Simplify $- 2.41812391 \cdot \log (25)$ by moving $2.41812391$ inside the logarithm.

$\frac{- \log (25^{2.41812391})}{\log (7)}$

Step 9

Simplify the numerator.

Raise $25$ to the power of $2.41812391$ .

$\frac{- \log (2400.\overline{9})}{\log (7)}$

Log base $10$ of $2400.\overline{9}$ is approximately $3.38039216$ .

$\frac{- 1 \cdot 3.38039216}{\log (7)}$

Step 10

Simplify the expression.

Multiply $- 1$ by $3.38039216$ .

$\frac{- 3.38039216}{\log (7)}$

Move the negative in front of the fraction.

$- \frac{3.38039216}{\log (7)}$

Step 11

Log base $10$ of $7$ is approximately $0.84509804$ .

$- \frac{3.38039216}{0.84509804}$

Step 12

Divide $3.38039216$ by $0.84509804$ .

$- 1 \cdot 4$

Step 13

Multiply $- 1$ by $4$ .

$- 4$

Evaluate log base 7 of 25* log base 0.2 of 49

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