Question

solve the equation.\
$10 \ln(100x) - 3 = 117$\
$x = ?$\
round to the nearest hundredth.

Explanation:

Step1: Isolate the logarithmic term

Add 3 to both sides of the equation:
$$10\ln(100x) - 3 + 3 = 117 + 3$$
$$10\ln(100x) = 120$$

Step2: Simplify the coefficient of the log

Divide both sides by 10:
$$\frac{10\ln(100x)}{10} = \frac{120}{10}$$
$$\ln(100x) = 12$$

Step3: Exponentiate both sides

Use the property that if $\ln(a)=b$, then $a = e^b$. So we have:
$$e^{\ln(100x)} = e^{12}$$
Since $e^{\ln(a)} = a$, this simplifies to:
$$100x = e^{12}$$

Step4: Solve for x

Divide both sides by 100:
$$x = \frac{e^{12}}{100}$$

Now we calculate the value of $e^{12}$. We know that $e\approx2.71828$, so $e^{12}\approx162754.7914$. Then:
$$x = \frac{162754.7914}{100} = 1627.547914$$

Step5: Round to the nearest hundredth

Rounding 1627.547914 to the nearest hundredth (two decimal places), we look at the third decimal place which is 7. Since 7 > 5, we round up the second decimal place. So 1627.547914 rounded to the nearest hundredth is 1627.55.

Answer:

\( x \approx 1627.55 \)