log3 .1+2log3 x=log3 2+log3 5 logx 5+log3 x=log3 10 logb 9=2 1/4log2 16+1/2logx 49=log2 x

Question

there are 4 questions separated by big spaces. if there is a space after a number or letter then the number or letter before it is little and to the bottom.

Answer 1

1. log₃0.1+2log₃x=log₃2+log₃5

2. logᵪ5+log₃x=log₃10

3. log[b]9=2

4. ¼log₂16+½logᵪ49=log₂x

1. log₃(0.1x²)=log₃10, so 0.1x²=10, x²=100, and x=10.

2. 1/log₅x=log₃(10/x)=log₅(10/x)/log₅3.

Let y=log₅x, 1/y=(log₅10-log₅x)/log₅3=(log₅10-y)/log₅3.

Cross-multiply: log₅3=ylog₅10-y², y²-ylog₅10+log₅3=0.

If we use the formula to solve the quadratic in y, we discover that the discriminant is negative, because 4log₅3>(log₅10)², so the solution is complex. There is no real solution for x.

3. Rewrite b²=9 so b=3.

4. We can evaluate and simplify: 1+logᵪ7=log₂x. logᵪ7=log₂(x/2).

Therefore 1/log₇x=log₂(x/2), and since log₇x=log₂x/log₂7, log₂7/log₂x=log₂x-1.

Let y=log₂x, log₂7/y=y-1, so y²-y-log₂7=0.

From this, using the quadratic formula y=(1±√(1+4log₂7))/2=2.2485 or -1.2485 approx.

x=2^y=4.7520 or 0.4209 approx.