Problem 19 - Algebra 2

Question

Use a table to solve each equation. Round to the nearest hundredth. $$ 6^{2 x}=10 $$

Step-by-Step Solution

Answer

The solution to the equation \(6^{2x} = 10\) when calculated and rounded to the nearest hundredth is x=0.36

Step 1: Set up the equation

Firstly, write down the given equation which is \(6^{2x}=10\).

Step 2: Apply logarithm on both sides

Apply logarithm to both sides of the equation. It is easier to work with if we use natural logarithm. This alters the equation to: \(\ln(6^{2x})=\ln(10)\)

Step 3: Use properties of logarithms

Use the fact that \(\ln(a^n) = n\ln(a)\). This changes the equation to: \(2x\ln(6)=\ln(10)\)

Step 4: Solve for 'x'

Isolate 'x' by dividing both sides of the equation by \(2\ln(6)\). This gives us: \(x = \frac{\ln(10)}{2\ln(6)}\)

Step 5: Calculate 'x' to the nearest hundredth

Now that we have the expression for 'x', we can substitute the values of \(\ln(10)\) and \(\ln(6)\) and round off the value of 'x' to the nearest hundredth. After making this calculation, 'x' comes out to be approximately 0.36.

Key Concepts

Natural Logarithm

The natural logarithm, often expressed as \( \ln \), is a special logarithm with a base of \( e \), where \( e \approx 2.71828 \). This constant \( e \) is a fundamental constant in math, similar to \( \pi \). Natural logarithms are particularly useful when dealing with exponential equations because they simplify the process of solving for variables in the exponent.

When you have an exponential equation with terms involving powers of \( e \), using natural logarithms helps in converting these into linear equations. This process transforms complex exponentiation into simpler multiplication, making it easier to isolate variables and solve the equation.

In our equation \( 6^{2x}=10 \), applying the natural logarithm to both sides gives us \( \ln(6^{2x})=\ln(10) \). This step is crucial to move forward with simplifying the equation using the properties of logarithms.

Properties of Logarithms

Logarithms are powerful mathematical tools that can simplify complex equations, especially when dealing with exponents. There are several properties of logarithms, but one of the most critical ones involves powers: \( \ln(a^b) = b \ln(a) \). This particular property is useful for breaking down equations where a variable is in the exponent.

By using this property, we can transform the equation from \( \ln(6^{2x}) \) into \( 2x \ln(6) \). Here, the exponent \( 2x \) comes down in front as a multiplier. This step is often referred to as "bringing the power down," and it significantly simplifies the process of solving for the variable \( x \).

Besides this, understanding other properties, such as the product, quotient, and change of base rule, can also aid in simplifying logarithmic expressions. However, in this scenario, the power property is the most pertinent.

Isolating Variables

Isolating the variable is usually the main goal when solving equations. After simplifying an equation with logarithms, the next step is to isolate the variable in question, so you can solve for its value.

In the equation \( 2x \ln(6) = \ln(10) \), isolating \( x \) involves dividing both sides by \( 2 \ln(6) \). This gives us the expression \( x = \frac{\ln(10)}{2 \ln(6)} \). By focusing on isolating \( x \), it allows us to calculate its precise value using known constants.

The act of isolating variables isn't exclusive to logarithmic equations. It is a foundational algebraic technique applied across various types of equations to find the value of unknowns. By mastering this skill, you can more easily tackle complex mathematical problems.