The logarithm of seven raised to the power of negative three times logarithm of five to base seven, to the base five is a logarithmic expression in arithmetic form, given in the math question. Now, let’s learn how to find the value of arithmetic logarithm expression by simplification.

The mathematical expression is formed by the combination of both logarithmic and exponential systems. It creates confusion in us while evaluating the expression and this type of expressions should be evaluated by reducing the complexity of the expression. Now, let us think about it logically.

The base number of the exponential expression inside the logarithmic expression is $7$ and the same base number is also base of logarithmic expression at exponent position. It creates an opportunity to eliminate the logarithmic expression from the exponent position.

$\implies$ $\log_{5}{7^{\displaystyle -3\log_{7}{5}}}$ $\,=\,$ $\log_{5}{7^{\displaystyle -3 \times \log_{7}{5}}}$

The factor $-3$ can be shifted to the exponent position of the number $5$ as per the power rule of logarithms.

$=\,\,$ $\log_{5}{7^{\displaystyle \log_{7}{5^{\displaystyle -3}}}}$

According to the fundamental rule of logarithms, the number $7$ raised to the power of logarithm of quantity to base $7$ is equal to the quantity. In this case, the quantity is the number $5$ is raised to the power of $-3$.

$=\,\,$ $\log_{5}{5^{\displaystyle -3}}$

Now, use the power rule of logarithms one more time to shift the exponent $-3$ as a factor in the expression.

$=\,\,$ $(-3) \times \log_{5}{5}$

The logarithm of $5$ to the base $5$ is equal to one as per the logarithm of base rule.

$=\,\,$ $(-3) \times 1$

Finally, multiply the factors $-3$ and $1$ to find the product and also to evaluate the given logarithmic expression.

$=\,\,$ $(-3)$

$=\,\,$ $-3$

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