How Can I Simplify This Logarithm?

Question

How can I simplify $\log_7 49^x$ ?

I realise that an online calculator can give me the answers, but I needed to learn the steps to arrive at the answer. I have now understood it - thank you.

As I don't have access to anybody in real life to assist me, the wonderful community here on this website is my only go to place for help.

I have realised too late in life that math is beautiful and am learning it all over again for the sake of wanting to love maths.

As stated before, the support and help I receive here is much appreciated and will continue to motivate me.

Answer 1

When in doubt... figure out what it means...

$\log_7 49^x = k$ means

$7^k = 49^x$

If you put $49$ to base $7$ that is $49^x = (7^2)^x = 7^{2x}$ so $k = 2x$.

Once you get comfortable with this you can have faith in "the rules".

$\log_b c^d = d\log_b c$ and $\log_b b^k = k$ so $\log_7 49^x = x\log_7 49=x\log_7 7^2 = x*2 = 2x$.

Answer 2

Note that

$$\log_7 49^x=x\log_7 49=2x$$

or as an alternative

$$\log_7 49^x=\log_7 7^{2x}=2x \log_7 7=2x$$

We have used that

$$\log_a b^c=c\log_a b$$

Answer 3

$$\log_7 49^x=x\log_7 49 =x\log_7 7^2$$ $$=2x\log_7 7 = 2x\cdot 1$$