Math

Question Find the value of aa that satisfies the equation 67=a6^{7}=a.

Studdy Solution

STEP 1

Assumptions
1. We have the equation 67=a6^{7}=a.
2. We are looking for an equivalent equation.

STEP 2

An equivalent equation is one that has the same solution set as the original equation. To find an equivalent equation, we can apply various algebraic manipulations as long as we do not change the value of the expression.

STEP 3

One way to create an equivalent equation is to take the logarithm of both sides of the equation. This will allow us to solve for aa in terms of a logarithmic expression.

STEP 4

Apply the logarithm to both sides of the equation. We can use the natural logarithm (denoted as ln\ln) or the common logarithm (denoted as log\log). For this example, we will use the natural logarithm.
ln(67)=ln(a)\ln(6^{7}) = \ln(a)

STEP 5

Using the power rule of logarithms, which states that ln(xy)=yln(x)\ln(x^{y}) = y \cdot \ln(x), we can simplify the left side of the equation.
7ln(6)=ln(a)7 \cdot \ln(6) = \ln(a)

STEP 6

The equation 7ln(6)=ln(a)7 \cdot \ln(6) = \ln(a) is equivalent to the original equation 67=a6^{7}=a. This is because taking the natural logarithm is a reversible operation, and we can exponentiate both sides to return to the original equation if needed.
The equivalent equation to 67=a6^{7}=a is 7ln(6)=ln(a)7 \cdot \ln(6) = \ln(a).

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