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Chapter 4: Problem 36

Solve logarithmic equation. \(4 x-24=\log _{x} 1 \quad(x>0, x \neq 1)\)

Short Answer

Expert verified
x = 6

Step by step solution

01

Rewrite the log equation

First, recognize that \(\text{log}_x 1 = 0\) for any valid base \( x\). So, the given equation becomes \(4x - 24 = 0\)
02

Solve for x

Next, solve the equation \(4x - 24 = 0\). Add 24 to both sides to get: \[4x = 24\]. Then, divide both sides by 4 to find \( x \, \ x = 6\).
03

Check the constraints

Verify the constraints \( x > 0 \) and \( x eq 1 \). Since \( x = 6\), it meets both constraints.

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Key Concepts

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

logarithms
Logarithms are mathematical operations that help us solve equations involving exponents. They are the inverse of exponentiation. Basically, if you have an equation of the form \( b^y = x \), you can rewrite it using a logarithm as \( \text{log}_b (x) = y \). This means that the logarithm of x with base b is equal to y.

In our exercise, we saw \( \text{log}_x 1 \). It's important to know that any number raised to the power of 0 is 1 (\( b^0 = 1 \)), which means \( \text{log}_x 1 \) is always 0 regardless of the base x. Understanding this property makes solving such logarithmic equations easier.
solving equations
Solving equations is all about isolating the variable you want to find. In the given exercise, after recognizing that \( \text{log}_x 1 = 0 \), we simplified the equation to \( 4x - 24 = 0 \). This is a straightforward linear equation.

Here are some general steps to solve such equations:
  • Bring constants to the other side by adding or subtracting them.
  • Divide or multiply both sides by the coefficient of the variable to isolate it.

Following these steps, we found that \( x = 6 \). Practicing these steps consistently helps improve problem-solving skills.
constraints
When solving equations, we must always check for constraints or conditions that the solutions must satisfy. Constraints ensure that the solution is valid for the original equation. For logarithmic equations, there are typically two main constraints: the base of the logarithm must be a positive number, and it cannot be 1.

In our exercise, we had the constraints \( x > 0 \) and \( x eq 1 \). These constraints are crucial because:
  • Logarithms are only defined for positive bases, and a base of 1 is not useful as \( \text{log}_1 x \) is undefined.
  • Checking the solution against these constraints ensures it is valid.

After solving, we found \( x = 6 \), which meets both constraints since 6 > 0 and 6 鈮� 1. Always remember to verify your solution against the problem's constraints.

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Most popular questions from this chapter

Concept Check Use properties of exponents to write each function in the form \(f(t)=k a^{\prime},\) where \(k\) is a constant. (Hint: Recall that \(a^{x+y}=a^{x} \cdot a^{y}\).) $$f(t)=3^{2 t+3}$$

If the function is one-to-one, find its inverse. $$\\{(-3,6),(2,1),(5,8)\\}$$

Evolution of Language The number of years, \(n\), since two independently evolving languages split off from a common ancestral language is approximated by $$ n \approx-7600 \log r $$ where \(r\) is the proportion of words from the ancestral language common to both languages. (a) Find \(n\) if \(r=0.9\) (b) Find \(n\) if \(r=0.3\) (c) How many years have elapsed since the split if half of the words of the ancestral language are common to both languages?

Use another type of logistic function. Tree Growth The height of a certain tree in feet after \(x\) years is modeled by $$ f(x)=\frac{50}{1+47.5 e^{-0.22 x}} $$ (a) Make a table for \(f\) starting at \(x=10,\) and incrementing by \(10 .\) What appears to be the maximum height of the tree? (b) Graph \(f\) and identify the horizontal asymptote. Explain its significance. (c) After how long was the tree 30 ft tall?

For each function as defined that is one-to-one, (a) write an equation for the inverse function in the form \(y=f^{-1}(x),\) (b) graph \(f\) and \(f^{-1}\) on the same axes, and \((c)\) give the domain and the range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$f(x)=\sqrt{6+x}, \quad x \geq-6$$
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