Which equation is equivalent to $\log _3(x+5)=2 ?$
$3^2=\left[\log _3(x+5)\right]^3$
$2^3=\left[\log _3(x+5)\right]^2$
$3^2=x+5$
$2^3=x+5$
Answer (1)
Use the definition of logarithm: if lo g b ( a ) = c , then b c = a .
Apply the definition to the given equation lo g 3 ( x + 5 ) = 2 , which gives 3 2 = x + 5 .
Compare the derived equation with the given options.
The equivalent equation is 3 2 = x + 5 .
Explanation
Understanding the Problem We are given the equation lo g 3 ( x + 5 ) = 2 and asked to find an equivalent equation from the given options.
Applying the Definition of Logarithm To find the equivalent equation, we use the definition of a logarithm: if lo g b ( a ) = c , then b c = a . In our case, b = 3 , a = x + 5 , and c = 2 .
Transforming the Equation Applying the definition, we get 3 2 = x + 5 .
Identifying the Matching Equation Now, we compare this derived equation with the given options:
3 2 = [ lo g 3 ( x + 5 ) ] 3
2 3 = [ lo g 3 ( x + 5 ) ] 2
3 2 = x + 5
2 3 = x + 5
The equation 3 2 = x + 5 matches the third option.
Final Answer Therefore, the equation equivalent to lo g 3 ( x + 5 ) = 2 is 3 2 = x + 5 .
Examples
Logarithmic equations are used in various fields such as calculating the magnitude of earthquakes on the Richter scale, determining the pH of a solution in chemistry, and modeling population growth in biology. Understanding how to convert between logarithmic and exponential forms is crucial in these applications. For example, if we know the hydrogen ion concentration in a solution, we can use logarithms to find the pH level, which indicates whether the solution is acidic or alkaline.
