Question

Solve for \(x\): \[ x+\log_{15}(5+3^x)=x\log_{15}5+\log_{15}24 \]

• A. \(2\)
• B. \(1\)
• C. \(5\)
• D. \(8\)

Hint:

Whenever a variable appears outside a logarithm, convert it into logarithmic form using: \[ x=\log_b(b^x) \] This helps combine all terms into a single logarithmic equation.

Answer 1

The Correct Option is B

Concept: To solve logarithmic equations, we first try to express all terms in logarithmic form with the same base. Then we use logarithmic identities such as: \[ \log_b m+\log_b n=\log_b(mn) \] and \[ a\log_b m=\log_b(m^a) \] After simplifying, the equation reduces to an algebraic or exponential equation.

Step 1: Converting the term \(x\) into logarithmic form.
Given: \[ x+\log_{15}(5+3^x)=x\log_{15}5+\log_{15}24 \] Since: \[ \log_{15}15=1 \] we can write: \[ x=x\log_{15}15 \] Using the power property of logarithms: \[ x\log_{15}15=\log_{15}(15^x) \] Thus, the equation becomes: \[ \log_{15}(15^x)+\log_{15}(5+3^x) = \log_{15}(5^x)+\log_{15}(24) \]

Step 2: Combining logarithms on both sides.
Using: \[ \log_b m+\log_b n=\log_b(mn) \] we get: \[ \log_{15}\left[15^x(5+3^x)\right] = \log_{15}(24\cdot5^x) \] Since logarithms with the same base are equal, their arguments must also be equal: \[ 15^x(5+3^x)=24\cdot5^x \]

Step 3: Simplifying the exponential equation.
Recall: \[ 15^x=5^x\cdot3^x \] Substitute this: \[ 5^x\cdot3^x(5+3^x)=24\cdot5^x \] Divide both sides by \(5^x\): \[ 3^x(5+3^x)=24 \]

Step 4: Reducing to a quadratic equation.
Let: \[ 3^x=t \] Then: \[ t(5+t)=24 \] Expanding: \[ t^2+5t=24 \] \[ t^2+5t-24=0 \] Factorizing: \[ (t+8)(t-3)=0 \] Thus, \[ t=-8 \quad \text{or} \quad t=3 \] Since: \[ t=3^x>0 \] the value \(t=-8\) is rejected. Therefore: \[ 3^x=3 \] \[ x=1 \]

Step 5: Verification of the solution.
Substitute \(x=1\) into the original equation: LHS: \[ 1+\log_{15}(5+3) = 1+\log_{15}(8) \] RHS: \[ 1\cdot\log_{15}5+\log_{15}24 \] Using logarithm property: \[ \log_{15}5+\log_{15}24 = \log_{15}(120) \] Also, \[ 1=\log_{15}15 \] Hence, \[ 1+\log_{15}8 = \log_{15}15+\log_{15}8 = \log_{15}(120) \] Both sides are equal, so the solution is correct. \[ \boxed{x=1} \]