Question(1 point) If you rewrite the expression $2 \log _{5} x-4 \log _{5}\left(x^{2}+1\right)+5 \log _{5}(x-1)$ as a single logarithm $\log _{5} A$ , then: $A=$ help (formulas)

Studdy Solution

STEP 1

1. The expression involves logarithms with the same base, which allows us to use logarithmic properties to combine them.
2. We will use properties of logarithms such as the power rule, product rule, and quotient rule to simplify the expression.

STEP 2

1. Apply the power rule to each logarithmic term.
2. Use the product and quotient rules to combine the terms into a single logarithm.

STEP 3

Apply the power rule to each logarithmic term:
$2 \log_{5} x = \log_{5} (x^2)$
$4 \log_{5} (x^2 + 1) = \log_{5} ((x^2 + 1)^4)$
$5 \log_{5} (x - 1) = \log_{5} ((x - 1)^5)$

STEP 4

Combine the terms using the product and quotient rules:
The expression becomes:
$\log_{5} (x^2) - \log_{5} ((x^2 + 1)^4) + \log_{5} ((x - 1)^5)$
Using the quotient rule for subtraction of logs:
$\log_{5} \left( \frac{x^2 \cdot (x - 1)^5}{(x^2 + 1)^4} \right)$

STEP 5

The expression is now a single logarithm:
$\log_{5} A$
Where:
$A = \frac{x^2 \cdot (x - 1)^5}{(x^2 + 1)^4}$
The value of $A$ is:
$\boxed{\frac{x^2 \cdot (x - 1)^5}{(x^2 + 1)^4}}$

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