Find the value of x that solves the equation ln(x-3)-ln 3=0. Answer:

Combine logarithms: To solve the equation ln(x-3) - ln 3 = 0, we can use the property of logarithms that allows us to combine the two logarithms into one by division since subtraction of logarithms corresponds to division of their arguments. So, we rewrite the equation as ln((x-3)/3) = 0.Exponentiate both sides: Next, we can exponentiate both sides of the equation to remove the natural logarithm. The equation e^(ln((x-3)/3)) = e^0 will simplify because e and ln are inverse functions. This gives us (x-3)/3 = e^0.Simplify the equation: Since e^0 is equal to 1, the equation simplifies to (x-3)/3 = 1.Isolate the term: To solve for x, we multiply both sides of the equation by 3 to isolate the term (x-3). This gives us x-3 = 3.Solve for x: Finally, we add 3 to both sides of the equation to solve for x. This gives us x = 3 + 3, which simplifies to x = 6.

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Q. Find the value of

x

that solves the equation

\ln (x-3)-\ln 3=0

\newline

Answer:

Combine logarithms: To solve the equation $\ln(x-3) - \ln 3 = 0$ , we can use the property of logarithms that allows us to combine the two logarithms into one by division since subtraction of logarithms corresponds to division of their arguments. $\newline$ So, we rewrite the equation as $\ln\left(\frac{x-3}{3}\right) = 0$ .
Exponentiate both sides: Next, we can exponentiate both sides of the equation to remove the natural logarithm. The equation $e^{\ln(\frac{x-3}{3})} = e^0$ will simplify because $e$ and $\ln$ are inverse functions. $\newline$ This gives us $\frac{x-3}{3} = e^0$ .
Simplify the equation: Since $e^0$ is equal to $1$ , the equation simplifies to $(x-3)/3 = 1$ .
Isolate the term: To solve for $x$ , we multiply both sides of the equation by $3$ to isolate the term $(x-3)$ . This gives us $x-3 = 3$ .
Solve for x: Finally, we add $3$ to both sides of the equation to solve for $x$ . This gives us $x = 3 + 3$ , which simplifies to $x = 6$ .