Question

How do you solve `Log_2 (10X + 4) - Log_2 X = 3`?

Answer 1

No solution.

Explanation:

First, simplify using the rule that `log_a b-log_a c=log_a(b/c)`.

`log_2((10X+4)/X)=3`

To undo the logarithm, exponentiate both sides with base `2`.

`2^(log_2((10X+4)/X))=2^3`

`(10X+4)/X=8`

Solve for `X`.

`10X+4=8X`

`X=-2`

Warning! This is an invalid answer. If `X=-2`, then both of the logarithm functions in the original equation would have a negative argument. It's impossible to take the logarithm of a negative number.

If we graph this as a function, the graph should never cross the `x`-axis, indicating a lack of roots.

graph{ln(10x+4)/ln2-lnx/ln2-3 [-5.64, 22.84, -3.47, 10.77]}