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Solve for the exact value of
x.

log_(5)(5x)+log_(5)(2)=0
Answer:

Solve for the exact value of $x$ . $\newline$ $\log _{5}(5 x)+\log _{5}(2)=0$ $\newline$ Answer:

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Product property of logarithms

Full solution

Q. Solve for the exact value of

x

.

\newline

\log _{5}(5 x)+\log _{5}(2)=0

\newline

Answer:

Apply Product Rule: Use the property of logarithms that allows us to combine the two logarithms into one by using the product rule. $\newline$ The product rule states that $\log_b(m) + \log_b(n) = \log_b(m*n)$ for any base $b$ . $\newline$ So, $\log_{5}(5x) + \log_{5}(2) = \log_{5}(5x*2)$ .
Simplify Expression: Simplify the expression inside the logarithm. $\newline$ $5x \times 2$ simplifies to $10x$ . $\newline$ So, $\log_{5}(5x) + \log_{5}(2) = \log_{5}(10x)$ .
Set Equal to Zero: Set the logarithmic expression equal to zero, as given in the equation. $\log_{5}(10x) = 0$ .
Convert to Exponential Form: Convert the logarithmic equation to its exponential form. $\newline$ The exponential form of $\log_b(a) = c$ is $b^c = a$ . $\newline$ So, $5^0 = 10x$ .
Solve for x: Solve for x. $\newline$ $5^0$ is equal to $1$ , so $1 = 10x$ . $\newline$ Divide both sides by $10$ to isolate $x$ . $\newline$ $x = \frac{1}{10}$ .

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