2^x=17
When bases are not the same, we can solve such equation by taking the logarithm of each side of the equation.
m=n ⇔ log m = log n
Note that in order to take their logarithms, both m and n must be positive numbers. We can see that it is the case.
b To solve the given logarithmic equation, we will rewrite it in exponential form using the definition of a logarithm.
log_b m= n ⇔ m= b^nThis definition tells us how to rewrite the logarithm equivalent of n in exponential form. The argument m is equal to b raised to the power of n.
log_3( x+1)=5 ⇔ x+1= 3^5
In our case, 5 is the exponent to which 3 must be raised to obtain x+1. Let's solve our equation.
c Examining the given equation, we can see that it contains a logarithm of a power with the same base.
log_3( 3^x)=4
This allows us to use the Inverse Property of Logarithms. Since a logarithmic function is the inverse of an exponential function, a logarithm and a power with the same base undo each other.
log_b( b^n)= n
This means that the logarithm on the left-hand side of the equation simplifies to x.
log_3( 3^x)=4 ⇒ x=4
Therefore, the solution to the equation is x=4.
d This time, we can see that the given equation contains a power with logarithm with the same base in the exponent.
4^(log_4(x))=7
Similarly as in Part C, we can use the Inverse Property of Logarithms. A power and a logarithm with the same base undo each other .
b^(log_b( a)) = a
In this case the power simplifies to x.
4^(log_4( x))=7 ⇒ x=7
We found that x=7.
