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Chapter 5: Problem 51

Convert to an exponential equation. \(\log _{a} M=-x\)

Short Answer

Expert verified
\[a^{-x} = M\]

Step by step solution

01

- Understand the Logarithmic Form

The given equation is in the form \(\log_{a} M = -x\). This means that \(a\) raised to some power results in \(M\).
02

- Recall the Exponential Form

The exponential form of a logarithmic equation \(\log_{a} M = -x\) is expressed as \[a^{\text{log}_{a} M} = M\]. Since \(\log_{a} M = -x\), we replace \(\log_{a} M\) with \(-x\).
03

- Convert to Exponential Form

Rewriting \(\log_{a} M = -x\) in exponential form gives \[a^{-x} = M\].

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Key Concepts

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

logarithmic form
Logarithms are a fundamental concept in mathematics. They answer the question: 'To what power must a base be raised, to produce a given number?'.
For example, the logarithmic form equation \(\text{\text{log}}_{a} M = -x\), breaks down into:
  • \text{a} - the base.
  • \text{M} - the result.
  • -x - the exponent or power.

An easy way to remember how this works is: A logarithm tells us how many of one number we multiply to get another number.
In this case, \(\text{\text{log}}_{a} M = -x\) means that \text{a} raised to the power of -x equals M.
So, if you ever find yourself confused, remember: logarithms are just another way to write exponents.
exponential form
The exponential form of a number is very straightforward. It indicates how many times a number (the base) is multiplied by itself to attain a certain result.
In our original example, \(\text{\text{log}}_{a} M = -x\), converting this to exponential form starts by recalling that \(\text{\text{log}}_{a} M\) essentially means the power to which we must raise \(\text{\text{a}}\) to get \(\text{\text{M}}\).
This means we describe \(\text{\text{a}}\) raised to the power of -x equal to M as \(\text{a}^{-x} = M\).
  • The base a,
  • is raised to the power of -x,
  • which equals M.
Thus, when you see an exponential equation, it directly shows you the process of repeatedly multiplying a base.
logarithmic to exponential conversion
Converting between logarithmic and exponential forms is a crucial skill in mathematics, especially in solving equations.
Here鈥檚 a step-by-step method for converting \(\text{\text{log}}_{a} M = -x\) to exponential form:
  • Step 1: Identify the logarithmic equation which is \(\text{\text{log}}_{a} M = -x\).
  • Step 2: Recognize that it means 'a raised to what power equals M?' Here, that power is -x.
  • Step 3: Write it as an exponent equation: \(\text{a}^{-x} = M\).

By understanding that logarithms are another way to express exponents, and because they鈥檙e inverses of each other, you become adept at flipping between these forms.
This versatility is essential, as some problems are more easily solved in their exponential form while others in their logarithmic form.
Practicing these conversions can significantly enhance your problem-solving ability in both algebra and calculus.

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Most popular questions from this chapter

Approximate the point \((s)\) of intersection of the pair of equations. $$y=2.3 \ln (x+10.7), y=10 e^{-0.007 x^{2}}$$

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