Write the expression below as a single logarithm in simplest form. 2log_(b)3-log_(b)9 Answer: log_(b)(◻)

Identify Properties: Identify the properties used to combine the logarithms. The expression given is 2log_(b)3 - log_(b)9. To combine these logarithms into a single logarithm, we will use the power property and the subtraction property of logarithms. Power Property: log_b(P^k) = k * log_b(P) Subtraction Property: log_b(P) - log_b(Q) = log_b(P/Q)Apply Power Property: Apply the power property to the first term. The first term is 2log_(b)3, which can be rewritten using the power property as log_(b)(3^2). So, 2log_(b)3 becomes log_(b)(3^2) or log_(b)(9).Combine Logarithms: Combine the two logarithms using the subtraction property. Now we have log_(b)(9) from the first term and -log_(b)9 from the second term. Using the subtraction property, we can combine these into a single logarithm: log_(b)(9) - log_(b)9 = log_(b)(9/9).Simplify Expression: Simplify the expression inside the logarithm. The expression inside the logarithm is 9/9, which simplifies to 1. So, log_(b)(9/9) becomes log_(b)(1).Recognize Identity: Recognize the logarithmic identity. The logarithm of 1 to any base is always 0. Therefore, log_(b)(1) = 0.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Write the expression below as a single logarithm in simplest form.

2log_(b)3-log_(b)9
Answer:
log_(b)(◻)

Write the expression below as a single logarithm in simplest form. $\newline$ $2 \log _{b} 3-\log _{b} 9$ $\newline$ Answer: $\log _{b}(\square)$

View step-by-step help

Home

Math Problems

Algebra 2

Quotient property of logarithms

Full solution

Q. Write the expression below as a single logarithm in simplest form.

\newline

2 \log _{b} 3-\log _{b} 9

\newline

Answer:

\log _{b}(\square)

Identify Properties: Identify the properties used to combine the logarithms. $\newline$ The expression given is $2\log_{b}3 - \log_{b}9$ . To combine these logarithms into a single logarithm, we will use the power property and the subtraction property of logarithms. $\newline$ Power Property: $\log_b(P^k) = k \cdot \log_b(P)$ $\newline$ Subtraction Property: $\log_b(P) - \log_b(Q) = \log_b\left(\frac{P}{Q}\right)$
Apply Power Property: Apply the power property to the first term. $\newline$ The first term is $2\log_{b}3$ , which can be rewritten using the power property as $\log_{b}(3^2)$ . $\newline$ So, $2\log_{b}3$ becomes $\log_{b}(3^2)$ or $\log_{b}(9)$ .
Combine Logarithms: Combine the two logarithms using the subtraction property. $\newline$ Now we have $\log_{b}(9)$ from the first term and $-\log_{b}9$ from the second term. Using the subtraction property, we can combine these into a single logarithm: $\newline$ $\log_{b}(9) - \log_{b}9 = \log_{b}(\frac{9}{9})$ .
Simplify Expression: Simplify the expression inside the logarithm. $\newline$ The expression inside the logarithm is $\frac{9}{9}$ , which simplifies to $1$ . $\newline$ So, $\log_{b}\left(\frac{9}{9}\right)$ becomes $\log_{b}(1)$ .
Recognize Identity: Recognize the logarithmic identity. $\newline$ The logarithm of $1$ to any base is always $0$ . $\newline$ Therefore, $\log_{b}(1) = 0$ .