Problem 22 - Precalculus

Question

In Exercises \(1-33,\) solve the equation analytically. $$ 25\left(\frac{4}{5}\right)^{x}=10 $$

Step-by-Step Solution

Answer

The solution is approximately \(x = 4.11\).

Step 1: Isolate the Exponential Expression

Start by isolating the exponential part of the equation. Divide both sides of 25\(\left(\frac{4}{5}\right)^{x}=10\) by 25:\[\left(\frac{4}{5}\right)^x = \frac{10}{25}\]Simplify the fraction on the right side:\[\left(\frac{4}{5}\right)^x = 0.4\]

Step 2: Convert to Logarithmic Form

To solve for \(x\), convert the equation \(\left(\frac{4}{5}\right)^x = 0.4\) to a logarithmic form:\[x = \log_{\left( \frac{4}{5} \right)}(0.4)\]

Step 3: Use Change of Base Formula

Use the change of base formula for logarithms to solve for \(x\). The change of base formula is:\[\log_a(b) = \frac{\log_c(b)}{\log_c(a)}\]Choosing base 10 for the logarithms, we have:\[x = \frac{\log(0.4)}{\log\left(\frac{4}{5}\right)}\]

Step 4: Calculate the Logarithms

Use a calculator to find the numerical values of the logarithms. Calculate:\[\log(0.4) \approx -0.39794\]\[\log\left(\frac{4}{5}\right) \approx -0.09691\]

Step 5: Divide the Logarithms

Divide the logarithm values obtained in Step 4:\[x \approx \frac{-0.39794}{-0.09691} \approx 4.106\]

Step 6: Round the Solution

Round the value of \(x\) to a reasonable number of decimal places, typically two for a simpler solution:\[x \approx 4.11\]

Key Concepts

Logarithmic Functions

Logarithmic functions are vital to understanding and solving exponential equations. They essentially do the reverse of what exponentials do. If you have an equation like \(a^x = b\), converting it to a logarithmic form \(x = \log_a(b)\) helps in easily finding the value of \(x\). The logarithm tells us what power we need to raise the base \(a\) to obtain \(b\).
In our exercise, we needed to solve \(\left(\frac{4}{5}\right)^x = 0.4\) by converting it to a logarithmic expression. This essentially turns the equation into a problem of finding which power of \(\frac{4}{5}\) results in 0.4.
Here's why logarithms are useful:

• They transform complicated multiplication and division into simpler addition and subtraction.
• They help solve exponential equations, which are otherwise tough to deal with directly.

So, by turning the exponential part into a logarithm, we have simplified our task dramatically.

Change of Base Formula

The change of base formula is a trick that makes it easier to compute logarithms with bases other than 10 or \(e\), which are the more familiar bases. This formula is expressed as:
\[\log_a(b) = \frac{\log_c(b)}{\log_c(a)}\]
In our step-by-step solution, we chose base 10 for simplicity since most calculators have a dedicated log button for it. The change of base allowed us to convert \(\log_{\left( \frac{4}{5} \right)}(0.4)\) into a manageable form, making it easier to compute using a calculator.
Why use the change of base formula?

• It allows conversion of any logarithm to a base that is more calculable with standard log functions.
• It opens up otherwise complex calculations to simpler inputs through a calculator.
• It enriches understanding of how different bases are related and expands the applicability of logarithms.

By applying this formula, we achieved the needed outcome from a seemingly tough computation.

Isolating Exponential Expressions

Isolating the exponential expression is an important first step in solving exponential equations. It involves rearranging the equation so that the exponential term stands alone on one side. This makes it straightforward to convert the expression into a logarithmic form.
In our exercise, the original equation was \(25\left(\frac{4}{5}\right)^x = 10\). To isolate \(\left(\frac{4}{5}\right)^x\), we divided both sides by 25. This left us with \(\left(\frac{4}{5}\right)^x = 0.4\). Without this isolation, moving forward to solve the equation becomes impossible.
Here are some reasons why isolating the exponential expression is critical:

• It provides a foundation to apply further mathematical techniques such as logarithms.
• It simplifies the equation, reducing the problem to one of finding the exponent.
• It is necessary for converting the equation into other mathematical forms, ensuring smooth transition from one solving method to another.

When we isolate the exponential part, we set ourselves up for successful solving of the equation using logarithmic methods.