Problem 15 - Algebra and Trigonometry

Question

Find the solution of the exponential equation, rounded to four decimal places. \(8^{0.4 x}=5\)

Step-by-Step Solution

Answer

The solution is approximately 1.9321.

Step 1: Understand the Exponential Equation

We are given the equation \(8^{0.4x} = 5\). Our task is to solve for \(x\). This equation involves an exponent containing the variable, which means we'll need to use logarithms to solve it.

Step 2: Take the Logarithm of Both Sides

To solve for \(x\), take the natural logarithm (or common logarithm) of both sides of the equation. This gives us:\[\ln(8^{0.4x}) = \ln(5)\]

Step 3: Apply the Logarithm Power Rule

Use the power rule for logarithms, which states \(\ln(a^b) = b \ln(a)\). Applying this rule, the equation becomes:\[0.4x \ln(8) = \ln(5)\]

Step 4: Solve for x

Now, solve for \(x\) by isolating it:\[0.4x = \frac{\ln(5)}{\ln(8)}\]Then divide both sides by 0.4:\[x = \frac{\ln(5)}{0.4 \ln(8)}\]

Step 5: Compute the Values

Calculate the natural logarithms and evaluate the expression using a calculator:- Compute \(\ln(5)\) and \(\ln(8)\)- Divide \(\ln(5)\) by the product of 0.4 and \(\ln(8)\)- Round the result to four decimal places.

Step 6: Present the Final Answer

Using a calculator, compute:\[x \approx \frac{1.6094}{0.4 \times 2.0794} \approx 1.9321\]Therefore, the value of \(x\), rounded to four decimal places, is approximately 1.9321.

Key Concepts

Logarithms

Logarithms are a powerful mathematical tool used to solve equations where the unknown variable is in the exponent. They can be thought of as the inverse operation of exponentiation.
To understand logarithms, consider the equation \(a^x = b\).

• The logarithm answers the question: What power of \(a\) will produce \(b\)?
• We write this as \(\log_a(b) = x\).

This conversion from exponential to logarithmic form simplifies many calculations. By using logarithms, complicated exponential equations become easier to handle since it allows us to "bring down" exponents so that they can be more easily solved.

Natural Logarithm

The natural logarithm is a specific type of logarithm that uses the mathematical constant \(e\) as its base, which is approximately 2.71828.
In mathematical notation, the natural logarithm of a number \(x\) is written as \(\ln(x)\).

• This base \(e\) is particularly important in calculus and many fields of science due to its unique properties.
• In practice, the natural logarithm is frequently used because it models many natural growth processes.

In the context of solving exponential equations, the natural logarithm helps simplify calculations because it corresponds with the base \(e\), making it a "natural" choice, particularly with continuous growth models and decay problems.

Power Rule for Logarithms

The power rule for logarithms is a useful property that simplifies expressions involving logarithms.
This rule states: \(\log_b(a^c) = c \log_b(a)\).

• It allows us to move a power out in front of the logarithm, effectively turning an exponent into a coefficient.
• This is particularly useful when solving exponential equations, as it helps to isolate variables within exponents.

In practical applications like our example \(\ln(8^{0.4x}) = \ln(5)\), the power rule transforms it into \(0.4x \ln(8) = \ln(5)\), making the equation much easier to solve for \(x\). Without the power rule, solving such equations would be much more complex and time-consuming.