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Chapter 4: Problem 7

Rewrite the following equations using exponents instead of logs. Estimate a solution for \(x\) and then check your estimate with a calculator. Round the value of \(x\) to the nearest integer. a. \(\log x=1.255\) b. \(\log x=3.51\) c. \(\log x=4.23\) d. \(\log x=7.65\)

Short Answer

Expert verified
a. 18 b. 3232 c. 16982 d. 446684

Step by step solution

01

Convert the logarithm to an exponential equation

Recall that \(\log_b(a) = c\) implies \[a = b^c\]. Since these are common logarithms (base 10), we have \(\log_{10}(x) = c\) implies \[x = 10^c\]. Apply this to each of the given equations.
02

Apply to part a

Given \(\log x = 1.255\), convert to exponential form: \[x = 10^{1.255}\]. Estimating this, we get \(\approx 18\). Use a calculator to verify: \[10^{1.255} \approx 18\]
03

Apply to part b

Given \(\log x = 3.51\), convert to exponential form: \[x = 10^{3.51}\]. Estimating this, we get \(\approx 3232\). Use a calculator to verify: \[10^{3.51} \approx 3232\]
04

Apply to part c

Given \(\log x = 4.23\), convert to exponential form: \[x = 10^{4.23}\]. Estimating this, we get \(\approx 16982\). Use a calculator to verify: \[10^{4.23} \approx 16982\]
05

Apply to part d

Given \(\log x = 7.65\), convert to exponential form: \[x = 10^{7.65}\]. Estimating this, we get \(\approx 446684\). Use a calculator to verify: \[10^{7.65} \approx 446684\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

exponential form
Understanding exponential form is essential when working with logarithms. When you see an equation like \(\text{log}_b(a) = c\), it can be converted into exponential form, which translates to \[a = b^c\]. This means \(a\) is the result you get when the base \(b\) is raised to the power of \(c\). For example, if you have \(\log_{10}(x) = 2\), converting it to exponential form gives \[x = 10^2\], which means \(x = 100\). This conversion is particularly useful to solve logarithmic equations and understand their relationship with exponential functions.
common logarithms
Common logarithms are logarithms with base 10, typically written as \(\log x\) instead of \(\log_{10} x\). When solving equations involving common logarithms, you simply convert from logarithmic to exponential form. For example, if given \(\log x = 3\), you convert it to \[x = 10^3\] to find that \(x = 1000\). The base of 10 is implied because common logarithms use base 10 by default. This common base makes calculations straightforward, particularly when using calculators that often have a \log\ button specifically for base 10 logarithms.
estimating solutions
To estimate solutions for logarithmic equations, you can convert them into exponential form and approximate the value. Suppose you have \(\log x=3.51\). Converting it to exponential form, you get \[x = 10^{3.51}\]. Since you know \(10^3 = 1000\) and \(10^4 = 10000\), you can estimate that \(x\) is somewhere in between. A closer estimate using a calculator would show that \(10^{3.51} \approx 3232\). Such estimation involves finding the power of ten that closely matches the exponent, allowing for a more intuitive grasp before confirming with a calculator.
exponential equations
Exponential equations have variables in their exponents. Solving them requires rewriting logarithmic forms into exponentials or vice versa. For instance, if you have \(\log x = 4.23\), you convert this directly into exponential form: \[x = 10^{4.23}\], which can be calculated to be about \(16982\). Recognizing exponential equations and converting them properly is crucial. It simplifies finding solutions, especially since exponential forms offer a clear pathway to numerical answers using properties of exponentiation.

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Most popular questions from this chapter

Verify that \(\left(\frac{2 a^{3}}{5 b^{2}}\right)^{4}=\frac{16 a^{12}}{625 b^{8}}\) using the rules of exponents.

Without using a calculator, find two consecutive integers such that one is smaller and one is larger than each of the following (for example, \(3<\sqrt{11}<4\) ). Show your reasoning. a. \(\sqrt{13}\) b. \(\sqrt{22}\) c. \(\sqrt{40}\)

Without using a calculator, show how you can solve for \(x\). a. \(10^{x-2}=100\) b. \(\log (x-4)=1\) c. \(10^{2 x-3}=1000\) d. \(\log (6-x)=-2\)

Radio waves, sent from a broadcast station and picked up by the antenna of your radio, are a form of electromagnetic (EM) radiation, as are microwaves, X-rays, and visible, infrared, and ultraviolet light. They all travel at the speed of light. Electromagnetic radiation can be thought of as oscillations like the vibrating strings of a violin or guitar or like ocean swells that have crests and troughs. The distance between the crest or peak of one wave and the next is called the wavelength. The number of times a wave crests per minute, or per second for fast-oscillating waves, is called its frequency. Wavelength and frequency are inversely proportional: the longer the wavelength, the lower the frequency, and vice versa-the faster the oscillation, the shorter the wavelength. For radio waves and other \(\mathrm{EM}\), the number of oscillations per second of a wave is measured in hertz, after the German scientist who first demonstrated that electrical waves could transmit information across space. One cycle or oscillation per second equals 1 hertz \((\mathrm{Hz})\). For the following exercise you may want to find an old radio or look on a stereo tuner at the AM and FM radio bands. You may see the notation \(\mathrm{kHz}\) beside the AM band and MHz beside the FM band. AM radio waves oscillate at frequencies measured in the kilohertz range, and FM radio waves oscillate at frequencies measured in the megahertz range. a. The Boston FM rock station WBCN transmits at \(104.1 \mathrm{MHz}\). Write its frequency in hertz using scientific notation. b. The Boston AM radio news station WBZ broadcasts at 1030 \(\mathrm{kHz}\). Write its frequency in hertz using scientific notation. The wavelength \(\lambda\) (Greek lambda) in meters and frequency \(\mu\) (Greek mu) in oscillations per second are related by the formula \(\lambda=\frac{c}{\mu}\) where \(c\) is the speed of light in meters per second. c. Estimate the wavelength of the WBCN FM radio transmission. d. Estimate the wavelength of the WBZ AM radio transmission. e. Compare your answers in parts (c) and (d), using orders of magnitude, with the length of a football field (approximately 100 meters).

Evaluate and write the result using scientific notation: a. \(\left(2.3 \cdot 10^{4}\right)\left(2.0 \cdot 10^{6}\right)\) b. \(\left(3.7 \cdot 10^{-5}\right)\left(1.1 \cdot 10^{8}\right)\) c. \(\frac{8.19 \cdot 10^{23}}{5.37 \cdot 10^{12}}\) d. \(\frac{3.25 \cdot 10^{8}}{6.29 \cdot 10^{15}}\) e. \(\left(6.2 \cdot 10^{52}\right)^{3}\) f. \(\left(5.1 \cdot 10^{-11}\right)^{2}\)
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