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Chapter 3: Problem 44

Find all numbers \(x\) such that the indicated equation holds. \(59=10^{3 x}\)

Short Answer

Expert verified
The solution to the given logarithmic equation \(\log_{4}(3x + 1) = -2\) is \(x = \frac{-5}{16}\).

Step by step solution

01

Identify the given logarithmic equation

We are given the equation: \[ \log_{4}(3x + 1) = -2 \]
02

Convert to exponential form

Recall that the logarithmic equation \(\log_{a}(b) = c\) can be rewritten in exponential form as \(a^c = b\). Using this, we can convert our given equation to exponential form: \[ 4^{-2} = 3x + 1 \]
03

Solve for x

First, simplify the left side of the equation: \[ \frac{1}{4^2} = 3x + 1 \] Now, subtract 1 from both sides of the equation: \[ \frac{1}{4^2} - 1 = 3x \] Find the common denominator and rewrite both fractions: \[ \frac{1 - 4^2}{4^2} = 3x \] Simplify the numerator: \[ \frac{1 - 16}{16} = 3x \] So we have: \[ \frac{-15}{16} = 3x \] Now, divide both sides of the equation by 3: \[ \frac{-15}{16\cdot3} = x \] Finally, we have the value of x: \[ x = \frac{-15}{48} \] Simplify the fraction: \[ x = \frac{-5}{16} \]
04

Conclusion

The solution to the given logarithmic equation \(\log_{4}(3x + 1) = -2\) is \(x = \frac{-5}{16}\).

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Key Concepts

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

Exponential Form
Understanding the exponential form is a fundamental concept in mathematics, especially when dealing with logarithmic equations. When we have an equation in the form
  • \(a^c = b\),
this is known as the exponential form. Here, \(a\) is the base, \(c\) is the exponent, and the result is \(b\). Let's break it down further:
  • The base is the number being multiplied by itself.
  • The exponent indicates how many times the base is used as a factor.
  • The result is the outcome of the base raised to the power of the exponent.
For example, in the equation \(4^{-2} = 3x + 1\), we see the base is 4 and the exponent is -2. The equation dictates multiplying 4 by itself -2 times, which also means taking the reciprocal as it's a negative exponent.
Logarithmic Form
The logarithmic form is another way of expressing an equation that can be converted into exponential form, enhancing our understanding and ability to solve equations. A logarithmic equation can be expressed as:
  • \(\log_{a}(b) = c\)
Here, \(a\) is the base, \(b\) is the result, and \(c\) is the exponent. It essentially asks, "To what power must \(a\) be raised, to produce \(b\)?" Let's see how this relates back to the exponential form:
  • The logarithmic form \(\log_{4}(3x + 1) = -2\) is equivalent to saying 4 raised to the power of -2 equals \(3x + 1\), or \(4^{-2} = 3x + 1\).
This transformation from logarithmic to exponential form makes it easier to solve the equation by dealing directly with an explicit power rather than a logarithm.
Solving for x
Once an equation is in exponential form, the next step is to solve it, finding the value for the variable, often labeled as \(x\). Let's go through the process simply:
  • The equation \(4^{-2} = 3x + 1\) can be rewritten to show the numerical value: \(\frac{1}{4^2} = 3x + 1\).
  • Simplify \(\frac{1}{16}\) and start isolating \(x\) by subtracting 1 from both sides of the equation. This gives:\( \frac{1}{16} - 1 = 3x\).
  • Convert both terms into fractions with a common denominator, simplifying it to \(\frac{-15}{16} = 3x\).
  • Finally, divide both sides by 3 to solve for \(x\): \(x = \frac{-15}{48}\), simplifying to \(x = \frac{-5}{16}\).
We've isolated \(x\) and calculated its value, showing that by understanding how to manipulate the equation in exponential form, finding unknowns becomes an achievable task.

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Most popular questions from this chapter

Find all numbers \(x\) that satisfy the given equation. \(\ln (x+4)-\ln (x-2)=3\)

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Find all numbers \(r\) such that \(\ln \left(2 r^{2}-3\right)=-1\).

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