Solve Logarithm Problem: Exponential Form & Single Expression

[math4life]

Homework Statement

1) Solve the equation using the exponential form of the equation.

Log₂(2-5x)=7

2) Rewrite as a single logarithmic expression:

3log x +(1/2)log z

Homework Equations

The Attempt at a Solution

1) I have no idea how to solve this.

2) I know that two logarithms multiply to add, but these have different bases so I do not know what to do.

Thanks.

 

[rock.freak667]

Log₂(2-5x)=7

Here is a hint.

a^b =c \Rightarrow b= log_a c

2) Rewrite as a single logarithmic expression:

3log x +(1/2)log z

they are in the same base. Logx usually means log₁₀x.

Use the rule

rlog_a x = log_a x^2

 

[Mentallic]

I'll just fix up rock.freak667's little typo there.

What was meant to be said is: r.log_a(x)=log_a(x^r)

Coupled with the rule that log_a(b)+log_a(c)=log_a(bc)

You should have no problem solving the second one

 

[Mark44]

I think it might be helpful to add that a logarithm can be thought of as the exponent on the base (the number raised to a power) that results in a particular number. So for example, log₁₀100 means the exponent on 10 that results in 100. In other words, this logarithm is the answer to the question 10^? = 100. Pretty clearly, the placeholder represented by ? is 2.

Every equation of the form log_bx = y can be rewritten as an equivalent exponential equation b^y = x, and vice versa, with the only restrictions being that b > 0, and b\neq 1, and x > 0.