Find the value of x that solves the equation ln(x+4)-(1)/(3)ln 27=ln 7. Answer: x=

Simplify using logarithm properties: First, we need to simplify the equation by using the properties of logarithms. The property we will use is that ln(a^b) = b*ln(a). We apply this to the term (1/3)ln 27.Rewrite and simplify: We know that 27 is 3^3, so we can rewrite (1/3)ln 27 as ln(27^(1/3)). Since 27^(1/3) is the cube root of 27, which is 3, this simplifies to ln(3).Combine logarithms: Now we have the equation ln(x+4) - ln(3) = ln(7). We can combine the logarithms on the left side of the equation using the property ln(a) - ln(b) = ln(a/b). This gives us ln((x+4)/3) = ln(7).Equating arguments: Since the natural logarithm function ln is one-to-one, we can equate the arguments of the logarithms to get (x+4)/3 = 7.Isolate x: To solve for x, we multiply both sides of the equation by 3 to isolate x+4. This gives us x+4 = 21.Solve for x: Finally, we subtract 4 from both sides to solve for x. This gives us x = 21 - 4, which simplifies to x = 17.

Home

Math Problems

Precalculus

Find the roots of factored polynomials

Full solution

Q. Find the value of

x

that solves the equation

\ln (x+4)-\frac{1}{3} \ln 27=\ln 7

\newline

Answer:

x=

Simplify using logarithm properties: First, we need to simplify the equation by using the properties of logarithms. The property we will use is that $\ln(a^b) = b\cdot\ln(a)$ . We apply this to the term $(\frac{1}{3})\ln 27$ .
Rewrite and simplify: We know that $27$ is $3^3$ , so we can rewrite $(1/3)\ln 27$ as $\ln(27^{1/3})$ . Since $27^{1/3}$ is the cube root of $27$ , which is $3$ , this simplifies to $\ln(3)$ .
Combine logarithms: Now we have the equation $\ln(x+4) - \ln(3) = \ln(7)$ . We can combine the logarithms on the left side of the equation using the property $\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$ . This gives us $\ln\left(\frac{x+4}{3}\right) = \ln(7)$ .
Equating arguments: Since the natural logarithm function $\ln$ is one-to-one, we can equate the arguments of the logarithms to get $\frac{x+4}{3} = 7$ .
Isolate $x$ : To solve for $x$ , we multiply both sides of the equation by $3$ to isolate $x+4$ . This gives us $x+4 = 21$ .
Solve for x: Finally, we subtract $4$ from both sides to solve for $x$ . This gives us $x = 21 - 4$ , which simplifies to $x = 17$ .