Problem 48
Question
For the following exercises, solve each equation for \(x\). \(\log _{8}(x+6)-\log _{8}(x)=\log _{8}(58)\)
Step-by-Step Solution
Verified Answer
The solution for \(x\) is \(\frac{2}{19}\).
1Step 1: Use the properties of logarithms
Recall the property of logarithms: \( \log_b (a) - \log_b (c) = \log_b \left( \frac{a}{c} \right) \). Applying this to the equation, we get: \( \log_{8} \left( \frac{x+6}{x} \right) = \log_{8}(58) \).
2Step 2: Set arguments equal
Since the logarithms on both sides have the same base, we can set their arguments equal: \( \frac{x+6}{x} = 58 \).
3Step 3: Solve the equation
Multiply both sides by \(x\) to eliminate the fraction: \(x + 6 = 58x\).
4Step 4: Rearrange the equation
Move \(x\) to the right side to start isolating \(x\): \(6 = 58x - x\).
5Step 5: Simplify and solve for x
Simplify the equation: \(6 = 57x\). Divide both sides by 57 to solve for \(x\): \(x = \frac{6}{57}\).
6Step 6: Simplify the fraction
Simplify the fraction: \(x = \frac{2}{19}\).
Key Concepts
Properties of LogarithmsSolving EquationsAlgebraic ManipulationFraction Simplification
Properties of Logarithms
Logarithms are mathematical functions that help us solve complex multiplications and powers. They have specific properties that make operations simpler. One of the key properties is the subtraction property, which states that:- If you have a logarithm subtraction, like \( \log_b (a) - \log_b (c) \), you can rewrite it as a single logarithm: \( \log_b \left( \frac{a}{c} \right) \).
This property is useful when solving equations, because it allows us to combine terms, making the problem simpler.
In our exercise, this property was applied to transform:- \( \log_{8}(x+6) - \log_{8}(x) = \log_{8}(58) \)
into
- \( \log_{8} \left( \frac{x+6}{x} \right) = \log_{8}(58) \).
This step is crucial as it reduces the equation to a simpler form to work with.
This property is useful when solving equations, because it allows us to combine terms, making the problem simpler.
In our exercise, this property was applied to transform:- \( \log_{8}(x+6) - \log_{8}(x) = \log_{8}(58) \)
into
- \( \log_{8} \left( \frac{x+6}{x} \right) = \log_{8}(58) \).
This step is crucial as it reduces the equation to a simpler form to work with.
Solving Equations
When solving logarithmic equations, after applying the logarithm properties, a crucial step is comparing the arguments.
In our equation, we had: \( \log_{8} \left( \frac{x+6}{x} \right) = \log_{8}(58) \).
Because the bases of the logarithms are the same (base 8 in this case), we can equate their arguments directly:- \( \frac{x+6}{x} = 58 \).
This step allows us to remove the logarithms from the equation and work with simpler algebraic expressions.
This method leverages the fact that if two logs with the same base are equal, then their arguments must also be equal.
This leads directly to a manageable algebraic equation.
In our equation, we had: \( \log_{8} \left( \frac{x+6}{x} \right) = \log_{8}(58) \).
Because the bases of the logarithms are the same (base 8 in this case), we can equate their arguments directly:- \( \frac{x+6}{x} = 58 \).
This step allows us to remove the logarithms from the equation and work with simpler algebraic expressions.
This method leverages the fact that if two logs with the same base are equal, then their arguments must also be equal.
This leads directly to a manageable algebraic equation.
Algebraic Manipulation
Algebraic manipulation is about using mathematical operations to simplify or rearrange equations.
Once we have \( \frac{x+6}{x} = 58 \), we can multiply through by \(x\) to eliminate the denominator:- \( x + 6 = 58x \).
This is a crucial step because fractions can often complicate solutions.
Once the denominators are cleared, we can focus on rearranging terms. By moving all terms involving \(x\) to one side, we get:- \( 6 = 58x - x \), which further simplifies to \( 6 = 57x \).
This isolating process helps focus on solving for the variable, making the solution clearer.
Once we have \( \frac{x+6}{x} = 58 \), we can multiply through by \(x\) to eliminate the denominator:- \( x + 6 = 58x \).
This is a crucial step because fractions can often complicate solutions.
Once the denominators are cleared, we can focus on rearranging terms. By moving all terms involving \(x\) to one side, we get:- \( 6 = 58x - x \), which further simplifies to \( 6 = 57x \).
This isolating process helps focus on solving for the variable, making the solution clearer.
Fraction Simplification
Fractions often need simplification to reach the final solution in an understandable form.
In our exercise, we reached the equation \( 6 = 57x \), which can be solved for \(x\) by dividing both sides by 57:- \( x = \frac{6}{57} \).
Simplifying fractions means reducing them to their simplest form. By finding the greatest common divisor, we simplify it to:- \( x = \frac{2}{19} \).
This makes the answer cleaner and easier to understand.
Fraction simplification is key in presenting solutions that are both accurate and elegant.
In our exercise, we reached the equation \( 6 = 57x \), which can be solved for \(x\) by dividing both sides by 57:- \( x = \frac{6}{57} \).
Simplifying fractions means reducing them to their simplest form. By finding the greatest common divisor, we simplify it to:- \( x = \frac{2}{19} \).
This makes the answer cleaner and easier to understand.
Fraction simplification is key in presenting solutions that are both accurate and elegant.
Other exercises in this chapter
Problem 47
For the following exercises, evaluate each function. Round answers to four decimal places, if necessary. \(f(x)=-2 e^{x-1},\) for \(f(-1)\)
View solution Problem 48
For the following exercises, find the value of the number shown on each logarithmic scale. Round all answers to the nearest thousandth. Plot each set of approxi
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For the following exercises, evaluate the common logarithmic expression without using a calculator. \(\log (1)+7\)
View solution Problem 48
For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth. \(12=2(3)^{x}+1\)
View solution