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What value of [tex]\( x \)[/tex] satisfies this equation?

[tex]\[ 10(1.5)^{3x} = 80 \][/tex]

Round your answer to the nearest hundredth.

The value of [tex]\( x \)[/tex] is [tex]\( \square \)[/tex]


Sagot :

To solve for [tex]\(x\)[/tex] in the equation [tex]\(10(1.5)^{3x} = 80\)[/tex], follow these steps:

1. Isolate the exponential term:
[tex]\[ (1.5)^{3x} = \frac{80}{10} \][/tex]
[tex]\[ (1.5)^{3x} = 8 \][/tex]

2. Take the natural logarithm of both sides:
[tex]\[ \ln\left((1.5)^{3x}\right) = \ln(8) \][/tex]

3. Use the properties of logarithms to bring the exponent down:
[tex]\[ 3x \cdot \ln(1.5) = \ln(8) \][/tex]

4. Solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{\ln(8)}{3 \cdot \ln(1.5)} \][/tex]

Upon calculating the values:
[tex]\[ \ln(8) \approx 2.0794 \][/tex]
[tex]\[ 3 \cdot \ln(1.5) \approx 1.2164 \][/tex]

5. Divide and round to the nearest hundredth:
[tex]\[ x = \frac{2.0794}{1.2164} \approx 1.71 \][/tex]

Therefore, the value of [tex]\(x\)[/tex] that satisfies the equation is:
[tex]\[ \boxed{1.71} \][/tex]