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Chapter 3: Problem 20

Evaluate the given quantities assuming that $$ \begin{array}{l} \log _{3} x=5.3 \text { and } \log _{3} y=2.1 \\ \log _{4} u=3.2 \text { and } \log _{4} v=1.3 \end{array} $$ $$ \log _{4}(2 u v) $$

Short Answer

Expert verified
The short answer for the given problem is: \(\log _{4}(2uv) = \log _{4}(2) + 4.5\)

Step by step solution

01

Determine the values of x, y, u, and v

We have: \(\log _{3}x = 5.3\) which means \(x = 3^{5.3}\) \(\log _{3}y = 2.1\) which means \(y = 3^{2.1}\) \(\log _{4}u = 3.2\) which means \(u = 4^{3.2}\) \(\log _{4}v = 1.3\) which means \(v = 4^{1.3}\) Now obtained the values of x, y, u, and v in terms of their respective bases.
02

Plug in the values of u and v into \(\log _{4}(2uv)\)

Now we need to calculate: \(\log _{4}(2uv) = \log _{4}(2(4^{3.2})(4^{1.3})) \)
03

Simplify using the logarithm properties

Using the logarithm properties, we can simplify the expression: \[\log _{4}(2 \cdot 4^{3.2} \cdot 4^{1.3}) = \log _{4}(2) + \log _{4}(4^{3.2}) + \log _{4}(4^{1.3}) \] Now, we know that \(\log _{4}4^{n} = n\), so we have: \[ \log _{4}(2) + 3.2 + 1.3 \]
04

Evaluate the expression

Now, we just need to evaluate the final expression: \[ \log _{4}(2) + 3.2 + 1.3 = \log _{4}(2) + 4.5 \] We cannot simplify this expression any further. Therefore, the final answer is: \[ \log _{4}(2uv) = \log _{4}(2) + 4.5 \]

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Most popular questions from this chapter

Suppose you have a calculator that can only compute square roots and can multiply. Explain how you could use this calculator to compute \(7^{3 / 4}\).

Evaluate the given quantities assuming that $$ \begin{array}{l} \log _{3} x=5.3 \text { and } \log _{3} y=2.1 \\ \log _{4} u=3.2 \text { and } \log _{4} v=1.3 \end{array} $$ $$ \log _{3} \frac{x}{3 y} $$

Find all numbers \(x\) that satisfy the given equation. $$ \log _{4}(x+4)-\log _{4}(x-2)=3 $$

Suppose a bank account paying \(4 \%\) interest per year, compounded 12 times per year, contains \(\$ 10,555\) at the end of 10 years. What was the initial amount deposited in the bank account?

Show that \(\sqrt{2+\sqrt{3}}=\sqrt{\frac{3}{2}}+\sqrt{\frac{1}{2}}\).
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