Solve for x natural log of 3+ natural log of 4x=4

Solution:

Step 1: Combine the logarithmic terms.

• Apply the product rule for logarithms: $\ln (a)+\ln (b)=\ln (ab)$.

• Combine the terms: $\ln (3)+\ln (4x)=\ln (3\cdot 4x)$.

Step 1.1: Simplify the combined logarithm.

• Multiply the constants: $\ln (12x)=4$.

Step 2: Convert the logarithmic equation to an exponential equation.

• Use the property of logarithms: $e^{\ln (a)}=a$.

• Apply this to both sides: $e^{\ln (12x)}=e^4$.

Step 3: Solve the exponential equation.

• Since $e^{\ln (a)}=a$, we have $12x=e^4$.

Step 4: Isolate the variable x.

• Divide both sides by 12 to solve for x: $x=\frac{e^4}{12}$.

Step 5: Present the final result.

• Exact form: $x=\frac{e^4}{12}$.

• Decimal form: $x\approx 4.54984583$ (rounded to the nearest ten-millionth).

Knowledge Notes:

To solve the given problem, we use several properties of logarithms and exponentials:

1. Product Rule for Logarithms: The logarithm of a product is the sum of the logarithms of the factors: $\ln (ab)=\ln (a)+\ln (b)$.

2. Exponential and Logarithmic Relationship: The natural logarithm $\ln (x)$ is the inverse function of the exponential function $e^x$. This means that $e^{\ln (x)}=x$ and $\ln (e^x)=x$.

3. Solving Logarithmic Equations: To solve equations involving logarithms, we often convert them into exponential form to isolate the variable.

4. Algebraic Manipulation: To isolate the variable, we perform algebraic operations such as multiplication, division, addition, and subtraction.

5. Exact and Decimal Forms: Solutions to equations can be represented in exact form (using symbols like $e$ for Euler's number) or in decimal form (an approximate value).

In this problem, we combined the logarithmic terms using the product rule, converted the logarithmic equation to an exponential equation, and then isolated the variable $x$ using algebraic manipulation. The final result was presented in both exact and decimal forms.