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Chapter 4: Problem 61

Write the logarithmic expression as a single logarithm with coefficient \(1,\) and simplify as much as possible. (See Exercises \(5-6)\) $$ 6 \log x-\frac{1}{3} \log y-\frac{2}{3} \log z $$

Short Answer

Expert verified
\log(x^6 y^{-1/3} z^{-2/3})

Step by step solution

01

Apply the Power Rule

Using the power rule of logarithms, which states that \[ a \, \log_b(x) = \log_b(x^a) \], we rewrite each term in the expression with the exponent: \[ 6 \, \log(x) = \log(x^6) \]. Similarly, \[ -\frac{1}{3} \, \log(y) = \log(y^{-\frac{1}{3}}) \]. Finally, \[ -\frac{2}{3} \, \log(z) = \log(z^{-\frac{2}{3}}) \].
02

Substitute and Combine the Logarithms

Now that we have rewritten each term using the power rule, substitute back into the original expression: \[ \log(x^6) + \log(y^{-\frac{1}{3}}) + \log(z^{-\frac{2}{3}}) \]. Next, use the product rule of logarithms, which states that \( \log_b(m) + \log_b(n) = \log_b(m \cdot n) \), to combine them into a single logarithm: \[ \log(x^6 \cdot y^{-\frac{1}{3}} \cdot z^{-\frac{2}{3}}) \].
03

Simplify the Expression

Simplify the expression inside the logarithm by combining the exponents: \[ x^6 \cdot y^{-\frac{1}{3}} \cdot z^{-\frac{2}{3}} \]. Since this is already in its simplest form, the final answer is: \[ \log(x^6 \cdot y^{-\frac{1}{3}} \cdot z^{-\frac{2}{3}}) \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithmic Functions
Logarithmic functions are the inverses of exponential functions. If you have an exponential function of the form \( y = b^x \), then the corresponding logarithmic function is \( x = \log_b(y) \). Logarithms represent the power to which a base, such as 10 or e, must be raised to produce a given number. For example, \( \log_2(8) = 3 \) because \( 2^3 = 8 \).
Power Rule of Logarithms
The power rule of logarithms allows us to handle exponents within logarithmic expressions. According to this rule, \( a \log_b(x) = \log_b(x^a) \).

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Most popular questions from this chapter

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(10^{5+8 x}+4200=84,000\)

Show that \(\ln \left(\frac{c+\sqrt{c^{2}-x^{2}}}{c-\sqrt{c^{2}-x^{2}}}\right)=2 \ln \left(c+\sqrt{c^{2}-x^{2}}\right)-2 \ln x\)

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(10^{3+4 x}-8100=120,000\)

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(\log _{6}(7 x-2)=1+\log _{6}(x+5)\)

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(\log _{3} y+\log _{3}(y+6)=3\)
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