Choose the single logarithmic expression that is equivalent to the one shown.
[tex]4 \log _3 x+\log _3 7[/tex]
[tex]\log _3 4 x^7[/tex]
[tex]\log _3 28 x[/tex]
[tex]\log _3 7 x^4[/tex]
[tex]\log _3\left(x^4+7\right)[/tex]
Answer (2)
Apply the power rule of logarithms: 4 lo g 3 x = lo g 3 x 4 .
Apply the product rule of logarithms: lo g 3 x 4 + lo g 3 7 = lo g 3 ( 7 x 4 ) .
The equivalent single logarithmic expression is lo g 3 ( 7 x 4 ) .
The correct option is lo g 3 7 x 4 .
Explanation
Understanding the problem We are given the expression 4 lo g 3 x + lo g 3 7 and asked to find an equivalent single logarithmic expression. We will use the properties of logarithms to combine these terms into a single logarithm.
Applying the power rule First, we use the power rule of logarithms, which states that a lo g b x = lo g b x a . Applying this to the first term, we get: 4 lo g 3 x = lo g 3 x 4
Applying the product rule Now our expression is lo g 3 x 4 + lo g 3 7 . We use the product rule of logarithms, which states that lo g b x + lo g b y = lo g b ( x y ) . Applying this rule, we have: lo g 3 x 4 + lo g 3 7 = lo g 3 ( 7 x 4 ) Thus, the equivalent single logarithmic expression is lo g 3 ( 7 x 4 ) .
Finding the correct option Comparing our result with the given options, we see that the equivalent expression is lo g 3 7 x 4 .
Examples
Logarithmic expressions are useful in many fields, such as calculating the magnitude of earthquakes on the Richter scale, determining the pH of a solution in chemistry, or modeling population growth in biology. For example, if you want to determine how much the intensity of an earthquake increased compared to a reference earthquake, you can use logarithmic scales to easily compare the magnitudes.
The expression 4 lo g 3 x + lo g 3 7 can be simplified to a single logarithm lo g 3 ( 7 x 4 ) by applying the power and product rules of logarithms. The correct answer from the options provided is lo g 3 7 x 4 .
;
