Question

Rewrite each equation in a different form. Solve for x. a) log100=x b) log₇1=x c) log(↓x)36=4 d) log₂x=-3 e) log₄1/16=x f) 2∧x=12

Answer 1

To rewrite and solve logarithmic equations, convert them to their exponential form, and vice versa for exponential equations. By manipulating the equations, you can isolate and solve for x, demonstrating the inverse relationship between logarithms and exponentiation.

Step-by-step explanation:

Rewriting Equations and Solving for x

To rewrite logarithmic equations into exponential form and solve for x, remember that logba=x is equivalent to b^x=a. For exponential equations, isolate x by using logarithms.

1. log10100=x becomes 10^x=100. Solving gives x=2.
2. log71=x becomes 7^x=1. All numbers raised to the 0th power equal 1, so x=0.
3. logx36=4 becomes x^4=36. Taking the fourth root gives x=√6.
4. log2x=-3 becomes 2^-3=x. The negative exponent means x=1/2^3, so x=1/8.
5. log41/16=x becomes 4^x=1/16. Because 1/16 is 4 to the negative 4th power, x=-4.
6. For 2^x=12, take the logarithm of both sides to solve. x=log(12)/log(2), or approximately x=3.58496.

These equations demonstrate relationships between logarithmic and exponential forms, and how to solve for the unknown variable, x, in different scenarios.