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

2log_(b)3-log_(b)3
Answer:
log_(b)(◻)

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

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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} 3

\newline

Answer:

\log _{b}(\square)

Understand and Identify: Understand the given expression and identify the logarithm properties to use. $\newline$ We have the expression $2\log_{b}3 - \log_{b}3$ . To combine these logarithms into a single logarithm, we can use the properties of logarithms, specifically the power rule and the subtraction rule. $\newline$ The power rule states that $n\log_{b}(a) = \log_{b}(a^n)$ . $\newline$ The subtraction rule states that $\log_{b}(a) - \log_{b}(c) = \log_{b}(a/c)$ .
Apply Power Rule: Apply the power rule to the first term of the expression. $\newline$ Using the power rule, we can rewrite $2\log_{b}3$ as $\log_{b}(3^2)$ . $\newline$ So, $2\log_{b}3$ becomes $\log_{b}(9)$ .
Combine Using Subtraction Rule: Combine the two logarithms using the subtraction rule. $\newline$ Now we have $\log_{b}(9) - \log_{b}(3)$ . Using the subtraction rule, we can combine these into a single logarithm: $\newline$ $\log_{b}(9) - \log_{b}(3) = \log_{b}\left(\frac{9}{3}\right)$ .
Simplify Fraction: Simplify the fraction inside the logarithm. $\newline$ Simplify $\frac{9}{3}$ to get $3$ . $\newline$ So, $\log_{b}\left(\frac{9}{3}\right)$ becomes $\log_{b}(3)$ .

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