Question

Solve the following exponential equation using logarithms.



−
4
e
(
7
a
+
9
)
+
6
=
−
6
−4e
(7a+9)
+6=−6



a
=
a=
(Round to the nearest hundredth

Answer 1

To solve the equation:

\[
-4e^{(7a + 9)} + 6 = -6,
\]

let's start by isolating the exponential term.

First, subtract 6 from both sides:

\[
-4e^{(7a + 9)} = -12.
\]

Next, divide both sides by -4:

\[
e^{(7a + 9)} = 3.
\]

Now, apply the natural logarithm to both sides:

\[
\ln(e^{(7a + 9)}) = \ln(3).
\]

Using the property of logarithms that states \(\ln(e^x) = x\), we have:

\[
7a + 9 = \ln(3).
\]

Now, let's solve for \(a\):

1. Subtract 9 from both sides:

\[
7a = \ln(3) - 9.
\]

2. Divide by 7:

\[
a = \frac{\ln(3) - 9}{7}.
\]

Now, let's calculate the value of \(a\):

1. Calculate \(\ln(3)\):

\[
\ln(3) \approx 1.0986.
\]

2. Substitute this value into the equation for \(a\):

\[
a \approx \frac{1.0986 - 9}{7} = \frac{-7.9014}{7} \approx -1.1288.
\]

Finally, rounding to the nearest hundredth:

\[
\boxed{-1.13}.
\]