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Chapter 8: Problem 15

Solve (a) \(\ln \left(\mathrm{e}^{x}\right)=5000\) (b) \(\ln \left(\mathrm{e}^{x}+10\right)=5\) (c) \(\frac{2}{3} \ln \left(x^{2}+9\right)=3\) (d) \(\log \left(\frac{x}{2}+1\right)=1.5\)

Short Answer

Expert verified
Question: Solve the following logarithmic equations: a) \(\ln(\mathrm{e}^{x}) = 5000\) b) \(\ln(\mathrm{e}^{x} + 10) = 5\) c) \(\frac{2}{3} \ln(x^2 + 9) = 3\) d) \(\log(\frac{x}{2} + 1) = 1.5\) Answer: a) \(x = 5000\) b) \(x = \ln(\mathrm{e}^{5} - 10)\) c) \(x = \pm\sqrt{\mathrm{e}^{\frac{9}{2}} - 9}\) d) \(x = 2(10^{1.5} - 1)\)

Step by step solution

01

Simplify the equation

The given equation is \(\ln(\mathrm{e}^{x}) = 5000\). We can simplify it using the fact that \(\ln(\mathrm{e}^{x}) = x\). So, we get \(x = 5000\). Result: \(x = 5000\) #Problem (b)#
02

Simplify the equation

The given equation is \(\ln(\mathrm{e}^{x} + 10) = 5\). We cannot directly apply logarithmic properties since the argument has a sum of two terms.
03

Solve for x

If we let \(y = \mathrm{e}^{x} + 10\), we can rewrite the equation as \(\ln(y) = 5\). To solve for \(y\), we can use the definition of natural logarithm: \(y = \mathrm{e}^{5}\). Then, substitute back into our previous definition of \(y\), \(\mathrm{e}^{x} + 10 = \mathrm{e}^{5}\), and solve for \(x\): \(x = \ln(\mathrm{e}^{5} - 10)\). Result: \(x = \ln(\mathrm{e}^{5} - 10)\) #Problem (c)#
04

Get rid of the fraction

The given equation is \(\frac{2}{3} \ln(x^2 + 9) = 3\). To eliminate the fraction, multiply both sides by 3: \(2 \ln(x^2 + 9) = 9\).
05

Eliminate the 2 before the logarithm

To remove the 2, divide both sides by 2: \(\ln(x^2 + 9) = \frac{9}{2}\).
06

Solve for x

Use the definition of natural logarithm to solve for the variable within the logarithm: \(x^2 + 9 = \mathrm{e}^{\frac{9}{2}}\). Solve for \(x^2\) as follows: \(x^2 = \mathrm{e}^{\frac{9}{2}} - 9\), and finally take the square root of both sides to find the possible values for \(x\): \(x = \pm\sqrt{\mathrm{e}^{\frac{9}{2}} - 9}\). Result: \(x = \pm\sqrt{\mathrm{e}^{\frac{9}{2}} - 9}\) #Problem (d)#
07

Rewrite the equation using exponential definition

The given equation is \(\log(\frac{x}{2} + 1) = 1.5\). We can rewrite it using the exponential definition (recalling that if \(\log_b{y} = x\), then \(y = b^x\)): \(\frac{x}{2} + 1 = 10^{1.5}\).
08

Solve for x

To solve for \(x\), rearrange the equation as follows: \(\frac{x}{2} = 10^{1.5} - 1\). Then multiply both sides by 2 to isolate \(x\): \(x = 2(10^{1.5} - 1)\). Result: \(x = 2(10^{1.5} - 1)\)

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Most popular questions from this chapter

Calculate the voltage gain in decibels of an amplifier where the input signal is \(0.15 \mathrm{~V}\) and the output signal is \(1.9 \mathrm{~V}\).

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