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Solve the exponential equation for [tex]\(x\)[/tex].

[tex]\[ 5^x = 20 \][/tex]

1. Exact Solution: [tex]\( x = \ \square \)[/tex]

2. Approximation (Round to 4 decimals): [tex]\( x = \ \square \)[/tex]


Sagot :

To solve the exponential equation [tex]\(5^x = 20\)[/tex], we can use logarithms to isolate the variable [tex]\(x\)[/tex]. Here are the detailed steps:

1. Taking the Logarithm of Both Sides:

We start by taking the logarithm of both sides with a common base. For convenience, we'll use the natural logarithm (logarithm base [tex]\(e\)[/tex]):

[tex]\[ \ln(5^x) = \ln(20) \][/tex]

2. Applying the Power Rule of Logarithms:

Using the logarithm power rule, [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex], we can move the exponent [tex]\(x\)[/tex] in front of the logarithm on the left-hand side of the equation:

[tex]\[ x \cdot \ln(5) = \ln(20) \][/tex]

3. Isolating [tex]\(x\)[/tex]:

Now, we solve for [tex]\(x\)[/tex] by dividing both sides of the equation by [tex]\(\ln(5)\)[/tex]:

[tex]\[ x = \frac{\ln(20)}{\ln(5)} \][/tex]

Thus, the exact solution for [tex]\(x\)[/tex] is:

[tex]\[ x = \frac{\ln(20)}{\ln(5)} \][/tex]

4. Finding the Numerical Value:

To find the numerical value, we can use a calculator to compute the natural logarithms:

- [tex]\(\ln(20) \approx 2.9957\)[/tex]
- [tex]\(\ln(5) \approx 1.6094\)[/tex]

Plugging these values into the equation, we get:

[tex]\[ x \approx \frac{2.9957}{1.6094} \approx 1.8613531161467862 \][/tex]

Therefore, the exact solution for [tex]\(x\)[/tex] is approximately [tex]\(1.8613531161467862\)[/tex].

5. Rounding for Approximation:

Finally, we can round this approximate value to four decimal places:

[tex]\[ x \approx 1.8614 \][/tex]

So, the solutions are:

1) Exact Solution:
[tex]\[ x = \frac{\ln(20)}{\ln(5)} = 1.8613531161467862 \][/tex]

2) Approximation (Rounded to 4 decimals):
[tex]\[ x = 1.8614 \][/tex]