a To solve the given logarithmic equation, we will rewrite it as an exponential equation using the definition of a logarithm.
y=log_b x ⇔ x= b^y
This definition tells us how to rewrite the logarithm equivalent of y as an exponential equation. The argument x is equal to b raised to the power of y.
x=log_(25)( 5) ⇔ 5= 25^x
We can see above that x is the exponent to which 25 must be raised to obtain 5. Now, let's solve the exponential equation by rewriting the terms so that they have a common base.
Now we have two equivalent expressions with the same base. If both sides of the equation are equal, the exponents must also be equal.
5^1=5^(2x) ⇔ 1=2x
Finally, we will solve the equation 1=2x.
b To solve the given logarithmic equation, we will rewrite it as an exponential equation using the definition of a logarithm.
log_b x=y ⇔ x= b^y
This definition tells us how to rewrite the logarithm equivalent of y as an exponential equation. The argument x is equal to b raised to the power of y.
log_x( 1)= ⇔ 1= x^()
Notice that any x raised to the power of is equal to 1. Since the logarithm base cannot equal 1, x can be any number except for 1.
c To solve the given logarithmic equation, we will rewrite it as an exponential equation using the definition of a logarithm.
y=log_b x ⇔ x= b^y
This definition tells us how to rewrite the logarithm equivalent of y as an exponential equation. The argument x is equal to b raised to the power of y.
23=log_(10)( x) ⇔ x= 10^(23)
