Question 1

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

Accepted Answer

$$6^3 = 216$

Question 2

Which of the following is the correct logarithmic form of $$8^{-\frac{2}$${3}} = \frac{1}{4}$$?

Accepted Answer

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

Question 3

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

Accepted Answer

$$3$$

Question 4

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

Accepted Answer

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

Question 5

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

Accepted Answer

$$-3$$

Question 6

Which of the following is the correct logarithmic form of $$10^{-3} = 0.001$?

Accepted Answer

$$\log_{10} 0.001 = -3$$

Question 7

Evaluate $$4^{\log_4 6}$$ without a calculator.

Accepted Answer

$$6$$

Question 8

A road construction crew is blasting through a mountain side. If the intensity of the blast is $$1.65 	imes 10$$^{-2}$$ W/m^2$, is this in violation of the 110 dB safety standard? Use D = 10$$ \log_{10} \left( \frac{I}{$$I_0$$} ight)$$, where I_0$ = 10$$^{-12}$$.

Accepted Answer

Yes, because $$D \approx 112.2 dB$$, which exceeds the standard.

Question 9

Solve for $$x in$$ $$10^x = 8000. $$Round your answer to the nearest thousandth.

Accepted Answer

$$x = 3.903$$

Question 10

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

Accepted Answer

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

Question 11

The decibel level $$D of a $$sound is given by $$D = 10 \log_{10} \left( \frac{I}{I_0$} \right). If$$ $$I = 10$$^{-12}$$ W/m^2$, what is D$?

Accepted Answer

$$0$$

Question 12

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

Accepted Answer

$$4.36$$

Question 13

Venus has a magnitude of $$-4.3$$ and Jupiter $$-2.1 on $$the astronomers’ scale. How many times brighter is Venus than Jupiter? Use $$m_2$$ - $$m_1 = -2.5$$ \log_{10} \left( \frac{$$b_2$$}{$$b_1$$} \right)$$.

Accepted Answer

$$8.32$$

Question 14

Evaluate $$\log_4$ 18 to $$four decimal places.

Accepted Answer

$$2.1699$$

Question 15

If $$\log_3$ 81 = 4$$, which of the following is true?

Accepted Answer

$$3^4 = 81$

Question 16

Evaluate $$\log_{0.8} 16 to $$four decimal places.

Accepted Answer

$$-8.9447$$

Question 17

The intensity levels $$L of $$two earthquakes can be compared by $$\log_{10} \left( \frac{I_1$}{I_2$} \right) = M_1$ - M_2$. If M_1$ = 9.0$$ and $$M_2 = 6.1$, how many times greater was the intensity of the 2011 earthquake? Round to the nearest hundredth.

Accepted Answer

$$794.33$$

Question 18

Express $$\log_{64} 8 = \frac{1}{2} in $$exponential form.

Accepted Answer

$$64^{1/2} = 8$

Question 19

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

Accepted Answer

$$-2$$

Logarithmic Expressions and Applications – Precalculus Study Guidance

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