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Chapter 3: Problem 106

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\log (3 x+4)=\log (x-10)$$

Short Answer

Expert verified
There is no solution to the given logarithmic equation.

Step by step solution

01

Utilize Logarithmic Properties

Using the properties of logarithms, set the arguments of the two logarithms equal to each other, because if \(\log(a) = \log(b)\), then \(a = b\). This gives us the equation \(3x + 4 = x - 10\).
02

Simplify The Equation

Move all terms involving \(x\) to one side of the equation and constants to the other side to isolate \(x\). You obtain this by subtracting \(x\) from both sides and subtracting 4 from both sides to get \(2x = -14\).
03

Solve For X

Finally, divide both sides of the equation by 2 to isolate \(x\). This yields the solution \(x = -7\). However, this solution must be checked back into the original equation to make sure it's within the domain of the logarithms (i.e., the arguments of the logarithms are positive).
04

Check the Solution

When substituting \(x = -7\) back into the original equation, both the arguments \(3x + 4\) and \(x - 10\) become negative, which is not possible in the logarithmic functions as the argument must be greater than zero. Thus, there is no solution to this problem.

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Most popular questions from this chapter

The table shows the time \(t\) (in seconds) required for a car to attain a speed of \(s\) miles per hour from a standing start. $$ \begin{array}{|c|c|} \hline \text { Speed, } s & \text { Time, } t \\ \hline 30 & 3.4 \\ 40 & 5.0 \\ 50 & 7.0 \\ 60 & 9.3 \\ 70 & 12.0 \\ 80 & 15.8 \\ 90 & 20.0 \\ \hline \end{array} $$ Two models for these data are as follows. \(t_{1}=40.757+0.556 s-15.817 \ln s\) \(t_{2}=1.2259+0.0023 s^{2}\) (a) Use the regression feature of a graphing utility to find a linear model \(t_{3}\) and an exponential model \(t_{4}\) for the data. (b) Use a graphing utility to graph the data and each model in the same viewing window. (c) Create a table comparing the data with estimates obtained from each model. (d) Use the results of part (c) to find the sum of the absolute values of the differences between the data and the estimated values given by each model. Based on the four sums, which model do you think best fits the data? Explain.

(a) solve for \(P\) and (b) solve for \(t\). $$A=P\left(1+\frac{r}{n}\right)^{n t}$$

A logistic growth model has the form ________.

A logarithmic model has the form ________ or ________.

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\ln x+\ln (x-2)=1$$
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