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Chapter 12: Problem 3

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log _{7}(7 x)$$

Short Answer

Expert verified
The expanded and simplified logarithmic expression is \(1 + \log_{7}(x)\).

Step by step solution

01

Identify the properties of logarithms

One of the properties of logarithms is the Product Rule, which stipulates that the log of a product is the sum of the logs. Meaning, \(\log _{b}(mn) = \log _{b}(m) + \log _{b}(n)\). This property will be applied in this exercise.
02

Apply the Product Rule

Use the Product Rule to expand \(\log _{7}(7x)\) into \(\log _{7}(7) + \log _{7}(x)\)
03

Simplify the expression

As per the definition of logarithms, if \(\log_{b}(a) = n\), then \(b^{n} = a\). Therefore, \(\log _{7}(7) = 1\) since \(7^{1}=7\). Substitute this into the expression to obtain \(1 + \log _{7}(x)\)

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Most popular questions from this chapter

Solve the system: $$\left\\{\begin{aligned}2 x &=11-5 y \\\3 x-2 y &=-12\end{aligned}\right.$$ (Section 4.3, Example 4)

Describe the relationship between an equation in logarithmic form and an equivalent equation in exponential form.

Use a graphing utility and the change-of-base property to graph \(y=\log _{3} x, y=\log _{25} x,\) and \(y=\log _{100} x\) in the same viewing rectangle. a. Which graph is on the top in the interval \((0,1) ?\) Which is on the bottom? b. Which graph is on the top in the interval \((1, \infty) ?\) Which is on the bottom? c. Generalize by writing a statement about which graph is on top, which is on the bottom, and in which intervals, using \(y=\log _{b} x\) where \(b>1\)

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$3^{x}=2 x+3$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(\log (x+3)=2,\) then \(e^{2}=x+3\)
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