Problem 86 - Algebra and Trigonometry

Question

In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ 10^{\log 53} $$

Step-by-Step Solution

Answer

The simplified expression of \(10^{log 53}\) is 53.

Step 1 - Identify the base

Identify the base of the exponent and the logarithm. We can see that both the logarithm and the exponent have the same base, which is 10.

Step 2 - Use the inverse property of logarithms

Using the inverse property of logarithms, \(a^{log_a b}=b\), that says if a number raised to the power of a logarithm with the same base results in the argument of the logarithm. The base 'a' is 10 in this equation, so we can simplify \(10^{log 53}\) to 53. This is essentially calculating 10 raised to the power that gives us 53, which is just 53 itself.

Key Concepts

Inverse Property of Logarithms

Logarithms and exponents are closely related mathematical concepts. The inverse property of logarithms is a key principle that greatly simplifies certain expressions. It's similar to undoing an operation, just like adding and subtracting are inverse operations. This property states that if you have a base raised to a logarithm of a number with the same base, you simply end up with that number. In mathematical terms, this is written as:

• \( a^{\log_a b} = b \)

In the context of our exercise, both the exponential base and the logarithmic base are 10. So, applying this inverse property yields

• \( 10^{\log 53} = 53 \).

This means the logarithm tells us exactly what power we need to raise 10 to get 53, which is why the expression simplifies straight down to 53. It essentially "cancels out" the logarithm.

Exponential Functions

Exponential functions are functions where a constant base is raised to a variable exponent. They are of the form:

• \( y = a^x \)

In these functions, the base \(a\) is positive, and the exponent \(x\) can be any real number. The base 10 exponentials, like the one in our exercise, are particularly important in logarithmic operations because they directly relate to common logarithms (logarithms with a base of 10).
Exponential functions often model situations of growth or decay, such as population growth or radioactive decay.
Using exponentials with logarithms, we can find the power needed to achieve a certain number, particularly when both use the same base.In the exercise, the base and the logarithm's base match, which informs how we apply the inverse property of logarithms. This interplay highlights why recognizing this relationship is crucial for simplification.

Simplification of Expressions

When we talk about simplifying expressions involving logarithms and exponentials, our goal is to make the expression as straightforward as possible.

• Look for opportunities to apply properties like the inverse property of logarithms, which can greatly shrink the apparent complexity of a problem.
• Check whether the base of the exponent and the base of the logarithm match. If they do, you might be able to use the inverse property.

Simplifying expressions not only makes solving problems easier, but it also deepens your understanding of their underlying properties.
Instead of blindly applying rules, it's important to see these shortcuts as tools to reveal simpler paths to your solution.
In our activity, the expression \(10^{\log 53}\) directly simplified to 53 using its properties, saving time and reducing the potential for errors in manual calculations.