To solve the equation [tex]\(2^{3x} = 10\)[/tex] and find an equivalent equation, let's follow these steps carefully:
1. Start with the given equation:
[tex]\[2^{3x} = 10\][/tex]
2. Take the logarithm base 2 on both sides of the equation:
[tex]\[\log_2(2^{3x}) = \log_2(10)\][/tex]
3. Apply the power rule of logarithms:
The power rule states that [tex]\(\log_b(a^c) = c \cdot \log_b(a)\)[/tex]. Using this rule:
[tex]\[\log_2(2^{3x}) = 3x \cdot \log_2(2)\][/tex]
4. Simplify the logarithm:
It is known that [tex]\(\log_2(2) = 1\)[/tex]. Therefore:
[tex]\[3x \cdot \log_2(2) = 3x \cdot 1 = 3x\][/tex]
5. Write the simplified left side of the equation:
[tex]\[3x = \log_2(10)\][/tex]
Therefore, the equivalent equation to [tex]\(2^{3x} = 10\)[/tex] is:
[tex]\[3x = \log_2(10)\][/tex]
This equation indicates that the value of [tex]\(3x\)[/tex] is equal to the logarithm base 2 of 10.
