Sure, let’s break it down step-by-step.
Given the logarithmic equation:
[tex]\[ \log_5 x - \log_5 25 = 7 \][/tex]
First, let's apply the properties of logarithms. One important property is:
[tex]\[ \log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right) \][/tex]
Applying this property to our given equation:
[tex]\[ \log_5\left(\frac{x}{25}\right) = 7 \][/tex]
This equation can be converted to its exponential form. The general rule for converting a logarithmic equation to an exponential form is:
[tex]\[ \log_b(a) = c \Rightarrow b^c = a \][/tex]
So, converting our equation:
[tex]\[ \log_5\left(\frac{x}{25}\right) = 7 \Rightarrow 5^7 = \frac{x}{25} \][/tex]
Next, we need to isolate [tex]\( x \)[/tex]:
[tex]\[ 5^7 = \frac{x}{25} \][/tex]
Multiplying both sides by 25 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = 25 \cdot 5^7 \][/tex]
Now, we recognize that 25 can be expressed as [tex]\( 5^2 \)[/tex]:
[tex]\[ x = 5^2 \cdot 5^7 \][/tex]
Using the properties of exponents [tex]\((a^m \cdot a^n = a^{m+n})\)[/tex]:
[tex]\[ x = 5^{2+7} \][/tex]
[tex]\[ x = 5^9 \][/tex]
Therefore, the exponential equation equivalent to the given logarithmic equation is:
[tex]\[ 5^9 = x \][/tex]
So, the correct answer is:
B. [tex]\( 5^9 = x \)[/tex]
