Solve the exponential equation. Use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution.

7^4x = 4.9

4x log7 = log4.9

x = log4.9/(4log7)
x = 0.20

To solve the exponential equation 7^(4x) = 4.9, we can take the logarithm of both sides. Since we want a decimal approximation, we'll use the natural logarithm (base e) or the logarithm with base 10.

Using the natural logarithm:
ln(7^(4x)) = ln(4.9)
Applying the property of logarithms, we can bring the exponent down:
4x * ln(7) = ln(4.9)
Now, divide both sides by ln(7):
4x = ln(4.9) / ln(7)
Using a calculator, compute the right hand side:
4x ≈ 0.603 / 1.9459 ≈ 0.310

Dividing both sides by 4:
x ≈ 0.310 / 4 ≈ 0.0775

Therefore, the decimal approximation for the solution is x ≈ 0.08 (rounded to two decimal places).

To solve the exponential equation 7^4x = 4.9, we need to find the value of x that satisfies the equation.

To solve for x, we can take the logarithm of both sides of the equation. In this case, we'll use the natural logarithm, which is denoted as ln.

Taking the natural logarithm of both sides gives us:

ln(7^4x) = ln(4.9)

Now, we can use the properties of logarithms to simplify the equation. The power rule of logarithms states that ln(a^b) = b * ln(a). Applying this rule, we have:

4x * ln(7) = ln(4.9)

Next, we'll isolate x by dividing both sides of the equation by 4ln(7):

x = ln(4.9)/(4 * ln(7))

Now, we can use a calculator to obtain a decimal approximation for the solution. Remember to use parentheses when entering the expression into the calculator to ensure the correct order of operations.

Using a calculator, we find:

x ≈ ln(4.9)/(4 * ln(7))

x ≈ -0.073

Therefore, the decimal approximation for the solution to the exponential equation 7^4x = 4.9, correct to two decimal places, is approximately -0.07.