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Evaluate the logarithmic expression without using a calculator. Remember that
log_(a)x is the exponent to which a must be raised in order to obtain
x.
(a)
log_(2)16
(d)
log_(2)sqrt2
(b)
log_(3)1
(e)
log_(e)((1)/(e^(2)))
(c)
log_(10)0.1
(f)
log_(1//2)8.

Evaluate the logarithmic expression without using a calculator. Remember that $\log _{a} x$ is the exponent to which a must be raised in order to obtain $\mathrm{x}$ . $\newline$ (a) $\log _{2} 16$ $\newline$ (d) $\log _{2} \sqrt{2}$ $\newline$ (b) $\log _{3} 1$ $\newline$ (e) $\log _{e}\left(\frac{1}{e^{2}}\right)$ $\newline$ (c) $\log _{10} 0.1$ $\newline$ (f) $\log _{1 / 2} 8$ .

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Quotient property of logarithms

Full solution

Q. Evaluate the logarithmic expression without using a calculator. Remember that

\log _{a} x

is the exponent to which a must be raised in order to obtain

\mathrm{x}

.

\newline

(a)

\log _{2} 16

\newline

(d)

\log _{2} \sqrt{2}

\newline

(b)

\log _{3} 1

\newline

(e)

\log _{e}\left(\frac{1}{e^{2}}\right)

\newline

(c)

\log _{10} 0.1

\newline

(f)

\log _{1 / 2} 8

.

Power of $2$ for $16$ : (a) $\log_{2}16$ $\newline$ Reasoning: $2$ to what power equals $16$ ? $\newline$ Calculation: $2^4 = 16$ $\newline$ Answer: $\log_{2}16 = 4$
Power of $2$ for $\sqrt{2}$ : (d) $\log_{2}\sqrt{2}$ $\newline$ Reasoning: $2$ to what power equals $\sqrt{2}$ ? $\newline$ Calculation: $2^{1/2} = \sqrt{2}$ $\newline$ Answer: $\log_{2}\sqrt{2} = \frac{1}{2}$
Power of $3$ for $1$ : (b) $\log_{3}1$ $\newline$ Reasoning: $3$ to what power equals $1$ ? $\newline$ Calculation: $3^0 = 1$ $\newline$ Answer: $\log_{3}1 = 0$
Power of e for $\frac{1}{e^2}$ : $\log_e\left(\frac{1}{e^{2}}\right)$ $\newline$ Reasoning: e to what power equals $\frac{1}{e^2}$ ? $\newline$ Calculation: $e^{-2} = \frac{1}{e^2}$ $\newline$ Answer: $\log_e\left(\frac{1}{e^{2}}\right) = -2$
Power of $10$ for $0$ . $1$ : $c$ $\log_{10}0.1$ $\newline$ Reasoning: $10$ to what power equals $0.1$ ? $\newline$ Calculation: $10^{-1} = 0.1$ $\newline$ Answer: $\log_{10}0.1 = -1$
Power of $1$ / $2$ for $8$ : $f$ $\log_{\frac{1}{2}}8$ $\newline$ Reasoning: $\left(\frac{1}{2}\right)^{?} = 8$ ? $\newline$ Calculation: $\left(\frac{1}{2}\right)^{-3} = 8$ $\newline$ Answer: $\log_{\frac{1}{2}}8 = -3$

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