The question brought up in the comments is deciding between whether your problem is [tex]8e^4x+8=15[/tex] ((8e^4)x) or [tex]8e^{4x}+8=15[/tex] ((8e^(4x)). There's also the third interpretation, [tex]8e^{4x+8}=15[/tex] (using parentheses, 8e^(4x+8)).
I'll assume it's the second of these:
[tex]8e^{4x}+8=15[/tex]
[tex]8e^{4x}=7[/tex]
[tex]e^{4x}=\dfrac78[/tex]
[tex]\ln(e^{4x})=\ln\dfrac78[/tex]
[tex]4x\ln e=\ln\dfrac78[/tex]
[tex]4x=\ln\dfrac78[/tex]
[tex]x=\dfrac14\ln\dfrac78[/tex]
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If instead it's the third possibility,
[tex]8e^{4x+8}=15[/tex]
[tex]e^{4x+8}=\dfrac{15}8[/tex]
[tex]\ln e^{4x+8}=\ln\dfrac{15}8[/tex]
[tex](4x+8)\ln e=\ln\dfrac{15}8[/tex]
[tex]4x+8=\ln\dfrac{15}8[/tex]
[tex]4x=\ln\dfrac{15}8-8[/tex]
[tex]x=\dfrac14\ln\dfrac{15}7-2[/tex]
