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Select the answer which is equivalent to the given expression using your calculator.

log_(343)49

(2)/(3)

(2)/(5)

(5)/(2)

(3)/(2)

Select the answer which is equivalent to the given expression using your calculator. $\newline$ $\log _{343} 49$ $\newline$ $\frac{2}{3}$ $\newline$ $\frac{2}{5}$ $\newline$ $\frac{5}{2}$ $\newline$ $\frac{3}{2}$

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Multiplication with rational exponents

Full solution

Q. Select the answer which is equivalent to the given expression using your calculator.

\newline

\log _{343} 49

\newline

\frac{2}{3}

\newline

\frac{2}{5}

\newline

\frac{5}{2}

\newline

\frac{3}{2}

Understand the problem: Understand the problem. $\newline$ We need to find the value of the logarithm $\log_{343}49$ . This means we are looking for the exponent that $343$ must be raised to in order to get $49$ .
Express as common base: Express $343$ and $49$ as powers of a common base. $\newline$ $343$ is $7^3$ and $49$ is $7^2$ . This is because $7 \times 7 \times 7 = 343$ and $7 \times 7 = 49$ .
Rewrite using new expressions: Rewrite the logarithm using the new expressions for $343$ and $49$ . $\newline$ $\log_{7^3}(7^2)$ is the new expression.
Apply power rule: Apply the logarithm power rule. $\newline$ The power rule of logarithms states that $\log_{b^m}(a^n) = \frac{n}{m} \cdot \log_b(a)$ . In our case, we can simplify $\log_{7^3}(7^2)$ to $\frac{2}{3} \cdot \log_7(7)$ .
Simplify the logarithm: Simplify the logarithm. $\newline$ Since $\log_7(7)$ is $1$ (because any number to the power of $1$ is itself), the expression simplifies to $\frac{2}{3} \times 1$ , which is just $\frac{2}{3}$ .

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