Proving a log by induction

Question

I had a test recently and there was a log question that was $$3\log_3(x) - 4\log_3(x) + 1/2\log_3(x).$$ When I solved it I got $$\log_3 \left(\frac{1}{\sqrt{x}}\right).$$ My teacher says that is incorrect, so I asked many people and they said I was correct. She said the only way I could get points back was if I prove it by induction, which I do not know what that is. I was wondering if my answer was right and if so how would I prove it by induction if thats possible? Thanks for all your help. Its critical I get these points back

Answer 1

Shifting the powers, you get:

$$ \log_3(x)^3 - \log_3(x)^4 + \log_3(x)^.5 $$ $$ = \log_3( x^3 / x^4) + \log_3(x)^.5 $$ $$ = \log_3 \frac 1 x + \log_3(x)^{0.5} $$ $$ = \log_3 \frac 1 x \cdot x^{0.5}$$ $$= \log_3 \frac 1 {x^{0.5}} == \text{CORRECT!} $$

Edit: Saw wrong question. OP answer was correct after all

Answer 2

$$ 3\log_3(x) - 4\log_3(x) + \frac 1 2 \log_3(x) $$

What is written above does not express any math problem, but if you had said that some words above it say "Write this as a single logarithm.", then you'd have a math problem to work on. (One of the flaws (or worse than flaws) of our system of coercing masses of people to learn topics in mathematics whose reason for inclusion in the curriculum is only that they're used in later subjects that most students never take is that they have these strange distorted ways of thinking (if "thinking" it may be called) in which students think that what is written above expresses a problem even when no words accompany it.)

Now notice that it says $$ 3 L - 4L + \frac 1 2 L. $$ You can do that because the three $\text{“}L\text{''}$s are all the same: all three are $\log_3 x$.

Next, the distributive law is $$ \left( 3 - 4 + \frac 1 2\right) L. $$ Arithmetic tells us that this is $$ \frac{-1}2 L. $$ All of that is done without knowing anything about logarithms. But the next step requires some knowledge of logarithms: $$ \frac{-1} 2 \log_3 x = \log_3 (x^{-1/2}). $$ If it had said "Write this expression as a single logarithm", then we're done with that.

Knowledge of exponents tells us that $x^{-1/2} = \dfrac 1 {\sqrt x}$. Hence the expression we're trying to write a just one logarithm is $$ \log_3 \frac 1 {\sqrt x}. $$

Induction is not an appropriate technique for any of this.