Question 1

Express $$\log_6$ 216 = 3 in $$exponential form.

Accepted Answer

$$6^3 = 216$

Question 2

Express $$8^{-\frac{2}$${3}} = \frac{1}{4}$$ in logarithmic form.

Accepted Answer

$$\log_8$ \frac{1}{4} = -\frac{2}{3}$$

Question 3

Evaluate $$\log_5$ 125$$ without a calculator.

Accepted Answer

$$3$$

Question 4

Evaluate $$\log_3$ \frac{1}{27}$$ without a calculator.

Accepted Answer

$$-3$$

Question 5

Which of the following is the correct exponential form for $$\log_{1/2} 32 = -5$$?

Accepted Answer

$$(\frac{1}{2}$$)^{-5}$$ = 32$$

Question 6

Evaluate $$\log_{25} 5$$ without a calculator.

Accepted Answer

$$\frac{1}{2}$$

Question 7

If $$10^x = 8000$$, what is $$x to $$the nearest thousandth?

Accepted Answer

$$x = 3.903$$

Question 8

A sound has intensity $$I = 10$$^{-12}$$ W/m^2$. Using D = 10$$ \log_{10} \frac{I}{$$I_0$$}$$ with I_0$ = 10$$^{-12}$$, what is the decibel level D$?

Accepted Answer

$$0$$

Question 9

A blasting charge has intensity $$1.65 	imes 10$$^{-2}$$ W/m^2$. Is this in violation of the 110 dB safety standard? Use D = 10$$ \log_{10} \frac{I}{$$I_0$$}$$, I_0$ = 10$$^{-12}$$.

Accepted Answer

Yes, because $$D \approx 122.2 dB$$

Question 10

Find the magnitude $$R of an $$earthquake where $$a = 325$$, $$T = 4$$, and $$B = 3.25$$ using $$R = \log_{10} \frac{a}{T} + B.$$

Accepted Answer

$$R \approx 5.36$$

Question 11

Find the magnitude $$R of an $$earthquake where $$a = 230$$, $$T = 2$$, and $$B = 4.5$$ using $$R = \log_{10} \frac{a}{T} + B.$$

Accepted Answer

$$R \approx 6.16$$

Question 12

Venus has a magnitude of $$-4.3$$ and Jupiter $$-2.1. $$How many times brighter is Venus than Jupiter? Use $$m_2$$ - $$m_1 = -2.5$$ \log_{10} \frac{$$b_2$$}{$$b_1$$}$$.

Accepted Answer

$$b_1$$/$$b_2$$ \approx 8.32$$

Question 13

In August 2009, Honshu, Japan registered a 6.1 magnitude quake. In March 2011, Fukushima had a 9.0 quake. How many times greater was the intensity of the 2011 earthquake? Use $$\log_{10} \frac{I_1$}{I_2$} = M_1$ - M_2$.

Accepted Answer

$$\approx 794.33$$ times greater

Question 14

Which of the following is the correct logarithmic form for $$81^{1/2} = 9$?

Accepted Answer

$$\log_{81} 9 = \frac{1}{2}$$

Question 15

Evaluate $$\log_3$ 81$$ without a calculator.

Accepted Answer

$$4$$

Question 16

Evaluate $$\log_3$ \frac{1}{9}$$ without a calculator.

Accepted Answer

$$-2$$

Question 17

If $$\log_4$ 18 is $$evaluated using a calculator, which value is closest to four decimal places?

Accepted Answer

$$2.1699$$

Question 18

If $$\log_{0.8} 16 is $$evaluated using a calculator, which value is closest to four decimal places?

Accepted Answer

$$-7.2171$$

Precalculus Logarithmic Expressions and Applications Study Guide

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