Question

Solve the equation analytically. (Analytically means with algebraic methods.) Write the answer in exact form using integers, fractions or bases raised to exponents, not decimals. Check for extraneous solutions and do not include extraneous solutions. Type DNE if there is no solution. log₅​(x)=log₁/₅​​(x)+6 x= (Note the base of the logarithm on the right is 1/5 )

Answer 1

To solve the equation analytically, we will use properties of logarithms. First, we can rewrite the equation as log₅​(x)=log₁/₅​​(x)+6. Since the bases are the same, the logarithmic expression will be equal if and only if the arguments are equal.

Step-by-step explanation:

To solve the equation analytically, we will use properties of logarithms. First, we can rewrite the equation as log₅​(x)=log₁/₅​​(x)+6. Since the bases are the same, the logarithmic expression will be equal if and only if the arguments are equal. Therefore, we have x = 1/5(x) + 6.

To simplify, we can multiply both sides of the equation by 5 to eliminate the fraction. This gives us 5x = x + 30.

Combining like terms, we have 4x = 30. Dividing both sides by 4, we find that x = 7.5. Therefore, the solution to the equation is x = 7.5.