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Chapter 3: Problem 24

Use the properties of logarithms to rewrite and simplify the logarithmic expression. $$\log _{2}\left(4^{2} \cdot 3^{4}\right)$$

Short Answer

Expert verified
The simplified form of the logarithmic expression is approximately 10.34.

Step by step solution

01

Expand Logarithm using Product Rule

The given problem is an application of the product rule of logarithms. The product rule says that the logarithm of the product of two numbers equals the sum of the logarithms of those numbers. So, we can rewrite the expression as follows:\n \( \log_{2}(4^{2}) + \log_{2}(3^{4}) \)
02

Apply Power Rule of Logarithms

The power rule says that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number. Therefore, we can continue the simplification process:\n \( 2 \cdot \log_{2}(4) + 4 \cdot \log_{2}(3) \)
03

Simplify the Logarithmic Expressions

Since \( \log_{a}(a) = 1 \), we can utilize this to further simplify the expression: \n \( 2 \cdot 2 + 4 \cdot \log_{2}(3) = 4 + 4 \cdot \log_{2}(3) \)
04

Final Approximation

For a more precise answer, using a calculator to approximate \( \log_{2}(3) \) will give us the final concise answer. Since \( \log_{2}(3) \) approximately equals to 1.585, the total sum will be: \n \( 4 + 4 \cdot 1.585 = 10.34 \) approximately.

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Most popular questions from this chapter

Is it possible for a logarithmic equation to have more than one extraneous solution? Explain.

Use the acidity model given by \(\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right],\) where acidity \((\mathrm{pH})\) is a measure of the hydrogen ion concentration \(\left[\mathrm{H}^{+}\right]\) (measured in moles of hydrogen per liter) of a solution. Find the \(\mathrm{pH}\) if \(\left[\mathrm{H}^{+}\right]=1.13 \times 10^{-5}\).

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

The yield \(V\) (in millions of cubic feet per acre) for a forest at age \(t\) years is given by \(V=6.7 e^{-48.1 / t}\) (a) Use a graphing utility to graph the function. (b) Determine the horizontal asymptote of the function. Interpret its meaning in the context of the problem. (c) Find the time necessary to obtain a yield of 1.3 million cubic feet.

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\log 4 x-\log (12+\sqrt{x})=2$$
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