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Chapter 13: Problem 41

Solve each logarithmic equation. $$\log _{6} h=-2$$

Short Answer

Expert verified
The short answer is: \(h = \frac{1}{36}\).

Step by step solution

01

Analyze the given equation

First, let us look carefully at the given equation: \(\log_{6} h = -2\) Our task is to find the value of h by solving this logarithmic equation.
02

Convert logarithmic equation to exponential form

To solve the equation, we need to change it from logarithmic form to exponential form, which will allow us to isolate and solve for h. Recall that the logarithmic equation \(\log_{a} b = c\) can be converted to exponential form as \(a^c = b\). Applying this property to our given equation, we get: \(6^{-2} = h\)
03

Solve for h

Now, we will compute the value of h by evaluating the exponential expression: \(h = 6^{-2} = \frac{1}{6^2} = \frac{1}{36}\) So, the value of h is \(\frac{1}{36}\).

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Key Concepts

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

Solving Logarithmic Equations
Solving logarithmic equations involves finding the unknown value that satisfies the equation when it has been expressed in the form of a logarithm. In the context of our given problem, the equation is \(\log_{6} h = -2\). Logarithmic equations can often appear daunting at first glance, but with a step-by-step approach, they can be tackled efficiently. The key to solving these equations is converting them into a form that is easier to manage, which typically involves transforming them into an exponential equation. This conversion allows for the use of basic algebraic manipulation to isolate and solve for the unknown variable, making the problem much more approachable.
Exponential Form
In the process of solving logarithmic equations, converting the equation into exponential form is a crucial step. The exponential form of an equation provides a clear pathway to finding the unknown value. To convert a logarithmic equation to exponential form, we use the identity: \(\log_{a} b = c\) becomes \(a^c = b\). This transformation is essential as it aligns the equation away from logarithmic complexities towards more familiar arithmetic operations.In the given exercise, \(\log_{6} h = -2\) becomes \(6^{-2} = h\) by applying this concept. This step changes the focus from a logarithmic equation to a simply calculable exponential equation, making it much easier to determine the value of \(h\) by simple evaluation of the power.
Logarithmic and Exponential Properties
Understanding the properties of logarithms and exponents is vital in solving logarithmic equations successfully. These properties help simplify and manipulate equations for easier computation.
  • Converting Logarithmic to Exponential: As discussed, the core property \(\log_{a} b = c\) transforming into \(a^c = b\) is fundamental for solving these equations.
  • Negative Exponents: In exponential form, \(6^{-2} = h\), understanding that negative exponents represent reciprocal values is crucial. The equation \(6^{-2}\) is simplified to \(\frac{1}{6^2}\), resulting in \(\frac{1}{36}\).
By leveraging these properties, we can seamlessly transition from the logarithmic complexity to simpler arithmetic, arriving at the correct solution.

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