To find the number of solutions to the equation
log4(x−1)=log2(x−3), we need to understand the relationship between the logarithmic bases.
We can use the change of base formula to rewrite the term on the left side in terms of base 2 logarithms. Recall that the change of base formula is:
logb(a)=‌Here, we want to rewrite
log4(x−1) in terms of base 2 . Since 4 is
22, we can express it as:
log4(x−1)=‌Since
log2(4)=log2(22)=2, we have:
log4(x−1)=‌Now our equation becomes:
‌=log2(x−3)To eliminate the fraction, we multiply both sides of the equation by 2 :
log2(x−1)=2log2(x−3)Using the power rule of logarithms, which states
alogb(c)=logb(ca), we can rewrite the right side of the equation:
log2(x−1)=log2((x−3)2)Since the bases are the same, we can set the arguments equal to each other:
x−1=(x−3)2Now, we solve the resulting quadratic equation. Expand the square on the right side:
x−1=x2−6x+9Rearrange all terms to one side of the equation to set it to zero:
x2−6x+9−x+1=0Simplify the equation:
x2−7x+10=0Now we solve this quadratic equation either by factoring, completing the square, or using the quadratic formula. This quadratic can be factored as:
(x−2)(x−5)=0Setting each factor equal to zero gives us the potential solutions:
x−2=0⟹x=2x−5=0⟹x=5However, these solutions must be checked to ensure they satisfy the original logarithmic equations. Both values must make the arguments of the logarithms positive:
For
x=2 :
x−1=1 (which is positive)
x−3=−1 (which is not positive; hence,
x=2 is not a valid solution)
For
x=5 :
x−1=4 (which is positive)
x−3=2 (which is positive; hence,
x=5 is a valid solution)