Solve for the exact value of x. log_(5)(4x)-2log_(5)(4)=1 Answer:

Apply Power Rule: Apply the power rule of logarithms to the term 2log_5(4). The power rule states that n*log_b(a) = log_b(a^n). Therefore, we can rewrite the equation as: log_5(4x) - log_5(4^2) = 1Simplify Term: Simplify the term 4^2. Calculating 4^2 gives us 16. So the equation now becomes: log_5(4x) - log_5(16) = 1Apply Quotient Rule: Apply the quotient rule of logarithms to combine the logarithms on the left side. The quotient rule states that log_b(a) - log_b(c) = log_b(a/c). Therefore, we can combine the logarithms: log_5(4x/16) = 1Simplify Fraction: Simplify the fraction inside the logarithm. Simplifying 4x/16 gives us x/4. So the equation now becomes: log_5(x/4) = 1Convert to Exponential Form: Convert the logarithmic equation to its exponential form. Using the definition of a logarithm, we can rewrite the equation as: 5^1 = x/4Solve for x: Solve for x. Multiplying both sides by 4 to isolate x gives us: x = 5 * 4 x = 20

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Solve for the exact value of
x.

log_(5)(4x)-2log_(5)(4)=1
Answer:

Solve for the exact value of $x$ . $\newline$ $\log _{5}(4 x)-2 \log _{5}(4)=1$ $\newline$ Answer:

View step-by-step help

Home

Math Problems

Precalculus

Product property of logarithms

Full solution

Q. Solve for the exact value of

x

\newline

\log _{5}(4 x)-2 \log _{5}(4)=1

\newline

Answer:

Apply Power Rule: Apply the power rule of logarithms to the term $2\log_5(4)$ . The power rule states that $n\log_b(a) = \log_b(a^n)$ . Therefore, we can rewrite the equation as: $\log_5(4^x) - \log_5(4^2) = 1$
Simplify Term: Simplify the term $4^2$ . Calculating $4^2$ gives us $16$ . So the equation now becomes: $\log_5(4x) - \log_5(16) = 1$
Apply Quotient Rule: Apply the quotient rule of logarithms to combine the logarithms on the left side. $\newline$ The quotient rule states that $\log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right)$ . Therefore, we can combine the logarithms: $\newline$ $\log_5\left(\frac{4x}{16}\right) = 1$
Simplify Fraction: Simplify the fraction inside the logarithm. $\newline$ Simplifying $\frac{4x}{16}$ gives us $\frac{x}{4}$ . So the equation now becomes: $\newline$ $\log_5\left(\frac{x}{4}\right) = 1$
Convert to Exponential Form: Convert the logarithmic equation to its exponential form. $\newline$ Using the definition of a logarithm, we can rewrite the equation as: $\newline$ $5^1 = \frac{x}{4}$
Solve for x: Solve for x. $\newline$ Multiplying both sides by $4$ to isolate $x$ gives us: $\newline$ $x = 5 \times 4$ $\newline$ $x = 20$