Problem 21 - Calculus and its applications

Question

Write an equivalent logarithmic equation. $$ 10^{-2}=0.01 $$

Step-by-Step Solution

Answer

The equivalent logarithmic equation is \(\log_{10}(0.01) = -2\).

Step 1: Identify the Base, Exponent, and Result

In the given equation, the term "base" refers to the number that is raised to a power, "exponent" refers to the power, and "result" is the outcome of that exponentiation. So for the equation \(10^{-2} = 0.01\), the base is 10, the exponent is -2, and the result is 0.01.

Step 2: Use Logarithmic Definition

The definition of a logarithm is that for a positive number, the logarithm base \(b\) of a positive number \(c\) is the exponent \(n\) to which \(b\) must be raised to produce \(c\). This can be written as \(b^n = c\) converts to \(\log_b(c) = n\).

Step 3: Apply Logarithmic Definition

By applying the definition from step 2, convert the equation \(10^{-2} = 0.01\) to its logarithmic form. Here, \(b = 10\), \(n = -2\), and \(c = 0.01\). So, the equivalent logarithmic equation will be \(\log_{10}(0.01) = -2\).

Key Concepts

Exponents

Exponents are fundamental components when dealing with powers in mathematics. When you see an equation like \(10^{-2} = 0.01\), the number 10 is known as the "base," and the number -2 is the "exponent." The base is the number that is being multiplied by itself, while the exponent tells you how many times to multiply the base by itself. If the exponent is negative, like in this scenario, it indicates the reciprocal of the base raised to the positive of that exponent.

Here's a simple breakdown:

• If \( a^n \), where \(a\) is the base and \(n\) is the exponent, then you multiply \(a\) by itself \(n\) times.
• For negative exponents, \( a^{-n} \) would mean \( \frac{1}{a^n} \). So \(10^{-2}\) translates to \(\frac{1}{10^2} = \frac{1}{100} = 0.01\).

Understanding exponents is crucial as they simplify the expression and calculations of large numbers using repeated multiplication or division.

Base of a Logarithm

The base of a logarithm is very similar to the base in exponents, but it's used in a slightly different context when dealing with logarithmic equations. For the equation \(10^{-2} = 0.01\), when converted to logarithmic form, the base remains consistent. Here, the base of the logarithm becomes that reference point we repeatedly raise, adjusting the exponent to find the result.

In logarithms:

• The base tells us the number we're considering to be repeatedly multiplied.
• For example, in \(\log_{10}(0.01) = -2\), 10 is the base of the logarithm.
• This means that 10 needs to be raised to the power of -2 to get the number 0.01.

Keeping track of the base is key as it determines the structure and calculation of your logarithmic operation.

Logarithmic Form

Logarithmic form is an essential concept when translating between exponents and logs. The idea is to express powers in terms of logs, which can sometimes simplify mathematical problems, especially with very large or very small numbers. For the equation \(10^{-2} = 0.01\), we convert it into its logarithmic form by identifying the base, the result, and the exponent.

In general:

• The exponential equation \(b^n = c\) can be rewritten in logarithmic form as \(\log_b(c) = n\).
• For our example, \(10^{-2} = 0.01\), the equivalent logarithmic form is \(\log_{10}(0.01) = -2\).
• What this tells us is that the base 10 raised to the power of -2 gives 0.01.

This conversion highlights the relationship between exponential growth or decay and logarithms, revealing how a log acts as an inverse operation to exponentiation.