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Chapter 10: Problem 335

In the following exercises, solve each equation. \(7 \log _{3} x=\log _{3} 128\)

Short Answer

Expert verified
x = \sqrt[7]{128} \

Step by step solution

01

Understand the equation

The given equation is \(7 \log_{3} x = \log_{3} 128\). We need to solve for \x\.
02

Use the properties of logarithms

Recall the property that allows us to equate the arguments when the logarithms have the same base: \( \log_b a = \log_b c \rightarrow a = c\). Apply this to the given equation.
03

Simplify the equation

By the properties of logarithms, we have \(7 \log_{3} x = \log_{3} 128\). This can be rewritten by isolating \log_{3} x\ as \( \log_{3} x = \frac{ \log_{3} 128 }{7}\).
04

Solve for x

To find x, we rewrite the equation using exponential form. This gives us \( x = 3^{ \frac{ \log_{3} 128 }{7} }\).
05

Simplify

Notice that \log_{3} 128\ is a constant. We can simplify this further, but for the final solution, we have \( x = \sqrt[7]{128} \).

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Key Concepts

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

Logarithmic Properties
Logarithmic properties are rules that help you simplify and solve logarithmic equations.
One essential property is that logarithms with the same base can be equated. This means if \(\log_b a = \log_b c\), then \(a = c\).
You can also use properties like the product rule, \(\log_b (mn) = \log_b m + \log_b n\), and the power rule, \(\log_b (m^p) = p \log_b m\).
These properties come in handy to break down complex logarithmic expressions into simpler terms, making it easier to solve equations.
Equating Arguments
When solving logarithmic equations, one effective technique is equating the arguments.
This works because if the logarithms have the same base, their arguments must be equal.
Take the equation \(7 \log_{3} x = \log_{3} 128\). Thanks to the property of logarithms that allows us to equate their arguments, we can transform the equation into \(7 \log_{3} x = \log_{3} 128 \rightarrow \log_{3} x = \frac{\log_{3} 128}{7}\).
This simplified equation now places us one step closer to solving for x.
Exponential Form Conversion
Exponential form conversion is a method where we transform a logarithmic equation into an exponential one.
Take the equation \(\log_3 x = \frac{\log_3 128}{7}\). To solve for x, we convert this logarithmic equation into its exponential form: \[x = 3^{\frac{\log_3 128}{7}}\].
By recognizing that \(\log_3 128\) is a constant, we can then simplify further if needed. In this case, simplifying leads us to the final solution: \(x = \sqrt[7]{128}\).
This approach often simplifies the task of solving logarithmic equations, as it transforms the problem into a more manageable form.

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Most popular questions from this chapter

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible. \(\log _{2} 80-\log _{2} 5\)

In the following exercises, use the Power Property of Logarithms to expand each. Simplify if possible. \(\log _{4} \sqrt{x}\)

In the following exercises, solve each equation. \(\log _{6} x+\log _{6}(x-5)=\log _{6} 24\)

In the following exercises, use the Power Property of Logarithms to expand each. Simplify if possible. \(\log x^{-2}\)

In the following exercises, solve for \(x\). \(\log x+\log (x-15)=2\)
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