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Chapter 12: Problem 27

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log _{b}\left(\frac{x^{2} y}{z^{2}}\right)$$

Short Answer

Expert verified
The expanded form of the expression \(\log _{b}\left(\frac{x^{2} y}{z^{2}}\right)\) is \(2\log_b(x) + \log_b(y) - 2\log_b(z)\).

Step by step solution

01

Apply Quotient Rule

Using the quotient rule which states that log(a/b) = log(a) - log(b), the given expression becomes:\n\n\[\log_b(x^2y) - \log_b(z^2)\]
02

Apply Product Rule

Now, apply the product rule, which says that log(ab) = log(a) + log(b). The expression now is:\n\n\[\log_b(x^2) + \log_b(y) - \log_b(z^2)\].
03

Apply Power Rule

Finally, apply the power rule, which states that log(a^n) = n log(a). The fully expanded expression becomes: \n\n\[2\log_b(x) + \log_b(y) - 2\log_b(z).\]

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Most popular questions from this chapter

I can solve \(4^{x}=15\) by writing the equation in logarithmic form.

$$\text { Solve: } x-2(3 x-2)>2 x-3$$

Because the equations \(2^{x}=15\) and \(2^{x}=16\) are similar, I solved them using the same method.

Complete the table for a savings account subject to continuous compounding ( \(A=P e^{n}\) ). Round answers to one decimal place. $$\begin{array}{l|c|l|c} \hline \begin{array}{l} \text { Amount } \\ \text { Invested } \end{array} & \begin{array}{l} \text { Annual Interest } \\ \text { Rate } \end{array} & \begin{array}{l} \text { Accumulated } \\ \text { Amount } \end{array} & \begin{array}{l} \text { Time } t \\ \text { in Years } \end{array} \\ \hline \$ 17,425 & 4.25 \% & \$ 25,000 & \\ \hline \end{array}$$

Solve each equation. $$5^{x^{2}-12}=25^{2 x}$$
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