Final answer:
To solve the equation 7^(-x-9)=4^(4x) using base 10, rewrite both sides with the same base. Then, equate the exponents and solve for x algebraically.
Step-by-step explanation:
To solve the equation 7^(-x-9)=4^(4x) using base 10, we need to rewrite both sides with the same base. In this case, we will rewrite the 7 and 4 as powers of 10, since we are using base 10. We know that 7 is equivalent to 10^log107 and 4 is equivalent to 10^log104. Using this, the equation becomes:
10^(-x-9 * log107) = 10^(4x * log104)
Since the bases are the same, we can equate the exponents:
-x-9 * log107 = 4x * log104
To solve for x, we can solve this equation algebraically:
-x * log107 - 9 * log107 = 4x * log104
Now we can combine like terms and isolate the variable:
3x * log107 = 9 * log107
Finally, divide both sides by 3 * log107:
x = 9 * log107 / 3 * log107