Respuesta :
The solution to the considered logarithmic equation is: x = 17
What is logarithm and some of its useful properties?
When you raise a number with an exponent, there comes a result.
Lets say you get
[tex]a^b = c[/tex]
Then, you can write 'b' in terms of 'a' and 'c' using logarithm as follows
[tex]b = log_a(c)[/tex]
Some properties of logarithm are:
[tex]\log_a(b) = \log_a(c) \implies b = c\\\\\log_a(b) + \log_a(c) = \log_a(b \times c)\\\\\log_a(b) - \log_a(c) = \log_a(\frac{b}{c})\\\\\log_a(b^c) = c \times \log_a(b)\\\\\log_b(b) = 1\\\\\log_a(b) + log_b(c) = \log_a(c)[/tex]
For this case, the equation given to us is:
[tex]\log_2(3x-1) = \log_4(x+8)\\[/tex]
We can use the property [tex]\log_a(b) + log_b(c) = \log_a(c)[/tex]
Adding [tex]\log_4(2)[/tex] on both side of the equation [tex]\log_2(3x-1) = \log_4(x+8)\\[/tex]
[tex]\log_4(2) + \log_2(3x-1) = \log_4(2) + \log_4(x+8)\\\\\log_4(3x -1) = \log_4(2 \times (x+8))\\\\3x - 1 = 2(x+8)\\3x - 1 = 2x +16\\x = 17[/tex]
Thus, the solution to the considered logarithmic equation is: x = 17
Learn more about logarithm here:
https://brainly.com/question/20835449
