Converting Logarithmic Equations to Base 2: A Step-by-Step Guide

[Physicsrapper]

How to solve this:

log(base16)x + log(base4)x + log(base2)x = 7

If I have log(base16)x for example and i make

10^(log(base16)x)

of it, can I transform the base 16 into an exponent?

It would look like that then:

10^(logx)^16 = x^16

Would that be correct?

Then, I could solve it:

10^(logx)^16 * 10^(logx)^4 * 10^(logx)^2 = 10^7

that would be

x^16 * x^4 * x^2 = 10^7

x^22 = 10^7
22 = log(base x)10^7

Is that correct? Can I solve it this way?

 

[ellipsis]

I don't (think) your method is valid, but I think using Latex in your post would help clarify your idea. Here's the normal way you tackle the problem:

$$
\log _{16}x+\log_{4}x+\log_{2}x=7
$$

If you ever calculated non-natural logs on your calculator, you know that...

$$
\log_{B}x = \frac{\ln x}{\ln B}
$$

Replace all left terms by their respective fraction:

$$
\frac{\ln x}{\ln 16}+\frac{\ln x}{\ln 4}+\frac{\ln x}{\ln 2}=7
$$

Recall that $\ln(a^b) = b\ln a$, and that $16$ and $4$ are powers of $2$.

$$
\frac{\ln x}{\ln 2^4}+\frac{\ln x}{\ln 2^2}+\frac{\ln x}{\ln 2}=7
$$
$$
\frac{\ln x}{4\ln 2}+\frac{\ln x}{2\ln 2}+\frac{\ln x}{\ln 2}=7
$$

Find the common denominator, and consolidate the expression into one fraction:

$$
\frac{\ln x}{4\ln 2}+\frac{2\ln x}{4\ln 2}+\frac{4\ln x}{4\ln 2}=7
$$
$$
\frac{7\ln x}{4\ln 2}=7
$$

Divide by $7$ to cancel them out, and multiply each side by $4\ln 2$ to isolate \ln x.

$$
\ln x=4\ln 2
$$

Recall that $4\ln 2 = \ln 2^4 = \ln 16$:

$$
\ln x = \ln 16.
$$

Raise both sides to e to cancel out the logarithms, and you find that:

$$
x = 16
$$

 

[Integral]

try converting all to log₂