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Chapter 3: Problem 51

Solve the exponential equation algebraically. Approximate the result to three decimal places. $$e^{3 x}=12$$

Short Answer

Expert verified
The solution to the equation is \(x \approx 0.845\).

Step by step solution

01

Applying Natural Logarithm

To solve for x, we need to get rid of \(e^{3x}\). We can do this by applying the natural logarithm on both sides of the equation: \(ln(e^{3x}) = ln(12)\). Now we use the property of logarithms that states \(ln(a^b) = b * ln(a)\), simplifying the equation to: \(3x * ln(e) = ln(12)\).
02

Simplify the Equation

Since the natural logarithm of e equals one (ln(e) = 1), the left side of the equation simplifies to: \(3x = ln(12)\)
03

Solve for x

Now that x is not part of an exponential any longer, we can solve for it by dividing both sides of the equation by 3: \(x = \frac{ln(12)}{3}\)
04

Compute x

In this stage we simply need to compute the right-hand side using a calculator or similar, to the mentioned precision point of three decimal places. Thus we get: \(x \approx 0.845\)

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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)

The number of bacteria in a culture is increasing according to the law of exponential growth. After 3 hours, there are 100 bacteria, and after 5 hours, there are 400 bacteria. How many bacteria will there be after 6 hours?

Use the Richter scale \(R=\log \frac{l}{I_{0}}\) for measuring the magnitudes of earthquakes. Find the magnitude \(R\) of each earthquake of intensity \(I\) (let \(I_{0}=1\) ). (a) \(I=199,500,000\) (b) \(I=48,275,000\) (c) \(I=17,000\)

A laptop computer that costs $$\$ 1150$$ new has a book value of $$\$ 550$$ after 2 years. (a) Find the linear model \(V=m t+b\). (b) Find the exponential model \(V=a e^{k t}\) (c) Use a graphing utility to graph the two models in the same viewing window. Which model depreciates faster in the first 2 years? (d) Find the book values of the computer after 1 year and after 3 years using each model. (e) Explain the advantages and disadvantages of using each model to a buyer and a seller.

Evaluate the logarithm using the change-of-base formula. Round your result to three decimal places. $$\log _{1 / 4} 5$$
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