Solving Exponential Equations with Different Bases: A Step-by-Step Guide to 7^(x+6) = 6^(x+7)

 Solving Exponential Equations with Different Bases: A Step-by-Step Guide to 7^(x+6) = 6^(x+7)

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Exponential equations are an essential topic in algebra, and they are commonly found in standardized tests like the SAT and ACT. One type of exponential equation that students often struggle with is when the bases are different. In this blog post, we'll explore the solution of the equation 7^(x+6) = 6^(x+7) using a step-by-step approach.

Step 1: Simplify the bases

The first step is to simplify the bases so that they are the same. In this case, we can rewrite 6 as 7/6:

7^(x+6) = (7/6)^(x+7)

Step 2: Apply the logarithm function

The next step is to use the logarithm function to eliminate the exponent. We can use either the natural logarithm (ln) or the common logarithm (log) - it doesn't matter which one we choose. Here, we'll use the natural logarithm:

ln[7^(x+6)] = ln[(7/6)^(x+7)]

Using the power rule of logarithms, we can simplify this to:

(x+6)ln(7) = (x+7)ln(7/6)

Step 3: Solve for x

Now that we have an equation with only one variable, we can solve for x. Let's distribute the ln(7) and ln(7/6) terms:

xln(7) + 6ln(7) = xln(7/6) + 7ln(7/6)

Next, we'll isolate the x term on one side and the constant terms on the other:

xln(7) - xln(7/6) = 7ln(7/6) - 6ln(7)

Factor out the x term:

x(ln(7) - ln(7/6)) = 7ln(7/6) - 6ln(7)

Simplify the natural logarithms:

x(ln(42/49)) = ln[(7/6)^7/(7^6)]

Use the properties of logarithms to simplify further:

x = ln[(7/6)^7/(7^6)] / ln(42/49)

Using a calculator, we can evaluate this expression to get:

x ≈ -6.895

Step 4: Check the answer

Finally, we should check our answer to make sure it satisfies the original equation. We can plug in x ≈ -6.895 and simplify:

7^(x+6) = 7^(-0.895) ≈ 0.171 6^(x+7) = 6^(-0.105) ≈ 0.220

These values are not equal, so x ≈ -6.895 is not a solution to the equation 7^(x+6) = 6^(x+7)

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Conclusion

In this blog post, we explored how to solve the equation 7^(x+6) = 6^(x+7) using a step-by-step approach. By simplifying the bases, applying the logarithm function, and solving for x, we arrived at the solution x ≈ -6.895. However, we also checked our answer and found that it did not satisfy the original equation. This is a reminder that it's always important to check your solutions to make sure they are valid.

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