QuestionSimplify $\frac{\log _{5} 25 \times \log _{22} 11}{\log _{3} 5}$ without tables or a calculator.

Studdy Solution

STEP 1

Assumptions1. We are dealing with logarithms, which are mathematical operations that help us find the power to which a certain base number must be raised to obtain another number. . We are using the properties of logarithms to simplify the expression.

STEP 2

First, we need to simplify the numerator of the fraction. We can use the property of logarithms that states $\log_{b} b^x = x$ .
$\log{5}25 \times \log{22}11 =2 \times1$

STEP 3

Now, we need to simplify the denominator of the fraction. We can use the property of logarithms that states $\log_{b} a = \frac{1}{\log_{a} b}$ .
$\log{3}5 = \frac{1}{\log_{5}3}$

STEP 4

Substitute the simplified numerator and denominator back into the original expression.
$\frac{2 \times1}{\frac{1}{\log_{}3}}$

STEP 5

implify the fraction by multiplying the numerator and denominator by $\log_{5}3$ .
$2 \times1 \times \log_{5}3$

STEP 6

implify the expression.
$2 \times \log_{5}3$ So, the simplified form of $\frac{\log{5}25 \times \log{22}11}{\log{3}5}$ is $2 \times \log_{5}3$ .

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