Replace the variables a and b with integers and solve for log a b = x by using the change of base formula. Show all work and use complete sentences to explain y

Question

Replace the variables a and b with integers and solve for log a b = x by using the change of base formula. Show all work and use complete sentences to explain your steps in solving the logarithm you created.

Please help!

Answer 1

Hey Angela,

 

log _a b = x        or     a^x  =  b

 

change -of-base formula will be:

 

log _a b= (log _n b) / (log _n a)    n is the new base.

 

 

 

Let's say 3^x = 7     or     log ₃ 7 = x

 

Take log both sides from first equation, and you will get  log 3^x = log 7.

Bring the x (exponent or power) to the front, and you will get: x (log 3) =  (log 7)

 

Then x = (log 7) / (log 3).  Remember: If there is no base, it will be base 10.  From this representation, 10 became new base.

 

 

Here is an example with integers in it:

 

So, let's go back to your equation log _a b = x

 

Let's rewrite as exponential equation:  a^x = b

 

Take log both sides and you will get   log a^x = log b.

 

Bring x (exponent or power) to the front and you will get    x (log a) = (log b).

 

 

Then, your x = (log b) / (log a)  (base for this log is 10)

 

Again, log_a b = x

 

Let a = 3, b = 81, x = 4          log₃81 = 4

 

log₁₀81 / log₁₀3 = 4

 

Rewrite ln function 81 with 3⁴ and 3 with 3¹:   log₁₀3⁴ /  log₁₀3¹ = 4

 

Guess what? you can bring those powers to the front and rewrite as

 

4 (log₁₀3) / 1 (log₁₀3)  =  4  (as log₁₀3 cancels out)

 

 

 

I'm going to use e as a base to make ln (easy to write)

 

log_e81 / log_e3   =   4        ln 81 / ln 3 = 4

 

Rewrite ln function 81 with 3⁴ and 3 with 3¹:   ln 3⁴ / ln 3¹ = 4

 

Guess what? you can bring those powers to the front and rewrite as

 

4 (ln 3) / 1 (ln 3) = 4   (as ln 3 cancels out)

 

 

 

I hope you understand / understood "Change-Base-Formula" from this.

 

Good luck to you.