To rewrite the function \( f(x) = 2 \cdot (5^{13x + 4})^3 - 7 \) in the form \( f(x) = a(b)^x - h + k \), we first simplify \( f(x) \).
We start by simplifying \( (5^{13x + 4})^3 \) using the property of exponents:
\[ (5^{13x + 4})^3 = 5^{3(13x + 4)} = 5^{39x + 12} \]
So, we can rewrite \( f(x) \) as:
\[ f(x) = 2 \cdot 5^{39x + 12} - 7 \]
Now, we want to express this in the form \( f(x) = a(b)^x - h + k \).
We can identify:
- \( a = 2 \)
- \( b = 5^{39} \) (since \( b^x \) would be \( 5^{39x} \), where \( b \) is effectively \( 5^{39} \))
- The term \( -h + k \) would then come from \( -7 \).
To express the function in the required form, we note the \( h \) term is related to the constant modifications:
-
The function becomes: \[ f(x) = 2 \cdot 5^{39x} \cdot 5^{12} - 7 \] Simplifying further, \[ f(x) = 2 \cdot 5^{39x} \cdot (5^{12}) - 7 \]
Recognizing \( k = 2 \cdot 5^{12} \) and noting we need it in the form \( -h + k \), we can express the \( -7 \) as \( -h + 0 \):
- Therefore, it implies \( -h = -7 \), giving \( h = 7 \).
Thus, the value of \( h \) is:
h = 7.