Answer :
The student is asking about solving an equation involving logarithms, exponential functions, and trigonometry. Let's break it down and solve it step-by-step.
Given:
[tex]\log_5 3 = t[/tex]
and
[tex]5^{2t-4} = 125[/tex]
We need to find the value of [tex]t[/tex].
Step 1: Solve for [tex]t[/tex] in the first equation.
The first equation is [tex]\log_5 3 = t[/tex]. This can be interpreted as:
[tex]5^t = 3[/tex]
So, [tex]t[/tex] is the power to which 5 must be raised to equal 3.
Step 2: Solve the second equation.
The second equation is [tex]5^{2t-4} = 125[/tex].
We know that 125 is a power of 5. Specifically,
[tex]125 = 5^3[/tex]
Substituting this, the equation becomes:
[tex]5^{2t-4} = 5^3[/tex]
Since the bases are the same, we can equate the exponents:
[tex]2t - 4 = 3[/tex]
Step 3: Solve for [tex]t[/tex].
Add 4 to both sides:
[tex]2t = 7[/tex]
Divide by 2 to solve for [tex]t[/tex]:
[tex]t = \frac{7}{2}[/tex]
Thus, the value of [tex]t[/tex] that satisfies both equations is [tex]\frac{7}{2}[/tex].