Problem 93 - Algebra 2

Question

Solve each equation. If necessary, round to the nearest ten-thousandth. $$ 4^{3 x}=77.2 $$

Step-by-Step Solution

Answer

\(x\)=0.7802

Step 1: Apply logarithms

Apply a natural logarithm (ln) to both sides of the equation. This gives us \(ln(4^{3 x})=ln(77.2)\).

Step 2: Use properties of logarithms

Use properties of logarithms to bring down the exponent: \(3 x \cdot ln(4)=ln(77.2)\).

Step 3: Isolate the variable

To isolate \(x\), divide both sides of the equation by \(3 ln(4)\). Giving \(x=\frac{ln(77.2)}{3 \cdot ln(4)}\).

Step 4: Compute the value

Now calculate the numerical value using a calculator. Please remember to round the final answer to the nearest ten-thousandth if necessary.

Key Concepts

Natural Logarithms

Natural logarithms are a fascinating and essential part of mathematics, especially when dealing with exponential equations. They are logarithms with the base "e," where "e" is approximately 2.71828. Unlike common logarithms that use "10" as their base, natural logarithms are often applied when working with natural growth patterns in science and mathematics.
To denote a natural logarithm, we use "ln", as seen in expressions like \( \ln(x) \). Acting as the inverse operation of exponential functions, they help simplify expressions involving exponents, making it easier to handle equations such as exponential growth or decay.
For instance, when solving the exponential equation \( 4^{3x} = 77.2 \), applying \( \ln \) to both sides helps manage the exponent, setting the stage for using more fundamental properties of logarithms.

Properties of Logarithms

The properties of logarithms are tools that make complex equations easier to handle. These properties transform challenging problems into more manageable forms.
One of the most crucial properties used in solving exponential equations is the "power rule," which states that \( \ln(a^b) = b \cdot \ln(a) \). This rule allows us to bring exponents down to the line in calculations, making it significantly simpler to isolate variables.
There's also the "product rule" \( \ln(ab) = \ln(a) + \ln(b) \) and the "quotient rule" \( \ln(\frac{a}{b}) = \ln(a) - \ln(b) \). Together, these properties provide flexibility in breaking down logarithmic expressions.
In our example with \( 4^{3x} = 77.2 \), we used the power rule to get \( 3x \cdot \ln(4) = \ln(77.2) \), which is crucial for simplifying the equation further.

Solving for Variables

Solving for variables is a key skill in mathematics, often requiring several steps to ensure that the variable is successfully isolated. It is a process of rearranging an equation so that the variable stands alone on one side of the equation sign.
In our equation \( 3x \cdot \ln(4) = \ln(77.2) \), we isolate \( x \) by dividing both sides by \( 3 \cdot \ln(4) \). This action rearranges the equation into \( x = \frac{\ln(77.2)}{3 \cdot \ln(4)} \).
The final step is computing the value of \( x \) using a calculator. It emphasizes precision, as numbers should often be rounded to a specified decimal place, like ten-thousandths here. By following these systematic steps, solving equations becomes a logical and orderly procedure.