Answer :
To rewrite the given log equation as an exponential equation, follow these steps:
1. Understand the given equation:
[tex]\[ \ln(3x + 9) = 2x - 3 \][/tex]
Here, [tex]\(\ln\)[/tex] denotes the natural logarithm, which is the logarithm to the base [tex]\(e\)[/tex].
2. Recall the property of logarithms:
The property that relates a logarithmic equation to an exponential equation is:
[tex]\[ \ln(a) = b \quad \text{if and only if} \quad e^b = a \][/tex]
3. Apply the property:
In the given equation [tex]\(\ln(3x + 9) = 2x - 3\)[/tex], let [tex]\(a = 3x + 9\)[/tex] and [tex]\(b = 2x - 3\)[/tex].
According to the property:
[tex]\[ \ln(3x + 9) = 2x - 3 \quad \text{can be rewritten as} \quad e^{2x - 3} = 3x + 9 \][/tex]
So, the exponential form of the given logarithmic equation is:
[tex]\[ e^{2x - 3} = 3x + 9 \][/tex]
1. Understand the given equation:
[tex]\[ \ln(3x + 9) = 2x - 3 \][/tex]
Here, [tex]\(\ln\)[/tex] denotes the natural logarithm, which is the logarithm to the base [tex]\(e\)[/tex].
2. Recall the property of logarithms:
The property that relates a logarithmic equation to an exponential equation is:
[tex]\[ \ln(a) = b \quad \text{if and only if} \quad e^b = a \][/tex]
3. Apply the property:
In the given equation [tex]\(\ln(3x + 9) = 2x - 3\)[/tex], let [tex]\(a = 3x + 9\)[/tex] and [tex]\(b = 2x - 3\)[/tex].
According to the property:
[tex]\[ \ln(3x + 9) = 2x - 3 \quad \text{can be rewritten as} \quad e^{2x - 3} = 3x + 9 \][/tex]
So, the exponential form of the given logarithmic equation is:
[tex]\[ e^{2x - 3} = 3x + 9 \][/tex]
