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Chapter 3: Problem 75

Condense the expression to the logarithm of a single quantity. $$\log x-2 \log (x+1)$$

Short Answer

Expert verified
The condensed form of the logarithmic expression is \(\log\frac{x}{(x+1)^2}\).

Step by step solution

01

Recognize the pattern

The given expression is a difference of two logarithms, hence we can use the property of logarithms states that the subtraction of logarithms is equivalent to the logarithm of the quotient: \(\log_ba - \log_bc = \log_b\frac{a}{c}\). Here, in the given expression, \(a = x\) and \(c = (x + 1)^2\) (since 2 in front shows it's squared due to power rule of log).
02

Apply the logarithmic subtraction property

By applying the logarithmic subtraction property, we change the difference of the logs into the log of a quotient. Therefore, the given expression \(\log x - 2\log(x+1) = \log \frac{x}{(x+1)^2}\).

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Most popular questions from this chapter

Use the following information for determining sound intensity. The level of sound \(\boldsymbol{\beta}\), in decibels, with an intensity of \(I\), is given by \(\boldsymbol{\beta}=10 \log \left(I / I_{0}\right),\) where \(I_{0}\) is an intensity of \(10^{-12}\) watt per square meter, corresponding roughly to the faintest sound that can be heard by the human ear. In Exercises 65 and 66 , find the level of sound \(\boldsymbol{\beta}\). (a) \(I=10^{-11}\) watt per \(\mathrm{m}^{2}\) (rustle of leaves) (b) \(I=10^{2}\) watt per \(\mathrm{m}^{2}\) (jet at 30 meters) (c) \(I=10^{-4}\) watt per \(\mathrm{m}^{2}\) (door slamming) (d) \(I=10^{-2}\) watt per \(\mathrm{m}^{2}\) (siren at 30 meters)

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\ln (x+5)=\ln (x-1)-\ln (x+1)$$

Use the acidity model given by \(\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right],\) where acidity \((\mathrm{pH})\) is a measure of the hydrogen ion concentration \(\left[\mathrm{H}^{+}\right]\) (measured in moles of hydrogen per liter) of a solution. Apple juice has a pH of 2.9 and drinking water has a pH of \(8.0 .\) The hydrogen ion concentration of the apple juice is how many times the concentration of drinking water?

Is it possible for a logarithmic equation to have more than one extraneous solution? Explain.

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$-x e^{-x}+e^{-x}=0$$
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